The question goes:

Which of the following is the set of all real numbers x such that x+3 > x+5?

(F)The empty set

(G)The set containing all real numbers

(H)The set containing all negative real numbers

(J)The set containing all nonnegative real numbers

(K)The set containing only zero

Alright, so as you probably guessed, students often don’t know what answer (F) means. First of all, let’s try to solve this problem and figure out the answer without thinking about the answer choices.

x+3 > x+5 (x on both sides, so maybe subtract x to get it by itself)

-x -x

3 > 5

Well that’s weird. Usually when we solve equations in math we end up with “x equals something” or “x is greater or less than something.” When we end up with something like 3>5 (or 5 = 6, or anything else that’s NOT true) the answer is **no solutions**. Definitely a good thing to know for the SAT and ACT (by the way, if you end up with something like 6 > 2 or 5=5 or something that IS true, the answer is **infinite solutions** or **all real numbers**).

Now, many of my students are familiar with this rule, and therefore know that this inequality has no solutions. If you get this far, you should *absolutely* be able to get the right answer here, even if you have no idea what answer (F) means.

If I look at the other answer choices, I am 100% sure that none of those things is the same thing as “no solution.” If you’re not sure, you could still *test* numbers in each set to see if they work. For example, if I wanted to test answer (K), I could just see if zero works as an answer. (0+3 is NOT greater than 0+5, so zero is NOT an answer here).

This type of situation often arises with vocab questions. Students know that 3 answers are wrong, and aren’t sure what the fourth word means. If that ever happens, PICK THE WORD YOU DON’T KNOW! Students often feel nervous doing that, but if you know the other answers are wrong, you should be confident picking the one you don’t understand.

Similarly here, even if you don’t know what “the empty set” means (by the way, it’s just another way of saying “no solutions” because the solution set is *empty*), you should definitely pick that if you know the other answers are wrong.

This strategy of picking an answer you don’t know if you’re sure that the other answers are wrong is definitely helpful, so don’t be afraid to use it!

Till next time!

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Even some of my strongest students tend to struggle with these questions, so I think it would be a good idea to go over what an “assumption” guiding scientific research might be, in the context of these reading comp passages.

The first example is from SAT practice test 3, where the last passage is talking about honeybees and some sort of mite problem they get (another page-turner from the Collegeboard).

I won’t make you read the whole passage just to follow along with this post, but the relevant part of the passage is as follows:

*This hypothesis can best be tested at trial, wherein a small portion of honeybees are offered a number of pyrethrum producing plants, as well as a typical bee food source such as clover, while controls are offered only the clover. Mites could then be introduced to each hive with note made as to choice of bees, and the effect of mite infestations on experimental groups and controls.*

And then question 49 asks:

An unstated assumption made by the authors about clover is that it

(a)does not produce pyrethrum

(b)are members of the *Chrysanthemum *genus

(c) are usually located near wild-type honeybee colonies

(d) will not be a good food source for honeybees in the control colonies

Ok, lets think about the experiment they’re proposing. Basically they want to test the effect that pyrethrum has on bees. And the way they’re going to do that is they’re going to give some bees pyrethrum producing plants AND clover, and the control group they’re only going to give clover.

Well, imagine you had to design an experiment to test whether pyrethrum has an effect on bees. How would you set it up? Most people would probably think “OK, let’s give some bees pyrethrum, and let’s not give some other bees pyrethrum, and see what the difference is.” Pretty reasonable right?

Seems like what our authors are assuming if they’re giving some bees only clover is that **clover does not have pyrethrum**. Right? Because if clover *also* has pyrethrum their experiment wouldn’t make any sense. It would be like saying “We’re going to test the effect that eating citrus fruit has on rats’ memory, so some rats we’re going to feed oranges AND lemons, but the control group we’re going to feed only lemons.” The problem with that experiment should be pretty obvious, right? We need our control group not to have the thing that we’re supposed to be testing, so this is what’s going on here.

Certainly not the easiest question, but the good news is that I’ve only seen two or three of these so far on the previous 8 SATs, and this one was the hardest.

**Also worth noting**, I think, is that even if ALL of this assumption business still doesn’t make sense to you, we should be able to *at the very least* eliminate answers (B) and (C) from the above question, as nowhere in that little excerpt about clover do we mention anything remotely comparable to Chrysanthemums or wild-type bee colonies. Always remember, with reading comp we don’t need to “figure out” the answers, we just need to *find *them in the passage.

Till next time!

]]>Today, I will once again veer off in a completely different direction and talk about one of math’s more annoying and frustrating concepts: inequalities.

Inequalities have always been a thorn in my side, and their presence can muddle even the most straightforward of questions.

First of all, a very brief review: inequality symbols show the relationships between numbers, variables, or a combination of the two. For example, when I say x > 5, I mean that x is **larger** than 5 (just remember, the alligator eats the bigger one). That means x could be 6, 7, 100, 5.00001, or anything else that is larger than 5. Conversely, if I say y < 7, that means that y is **smaller** than 7 (2, 4, -100, 6.9999, or anything else less than 7).

Easy enough, right?

Unfortunately, not really.

A big problem with inequalities is they throw people off and make otherwise easy questions seem more difficult. Imagine the following problem.

4(2x-5) > 6x + 2

Most of my students would have little trouble solving this equation if there were an equal sign in the middle. Therefore, the biggest piece of advice I give students (and something that I do myself) is to **imagine the inequality symbol is an equal sign and just solve it normally.**

Ok, so normally with equations like the one above, we want to get x by itself, so lets just try to do that!

8x – 20 > 6x + 2 (Distributive property)

2x – 20 > 2 (Subtract 6x from both sides)

2x > 22 (Add 20 to both sides)

x > 11 (Divide both sides by 2)

Really not so terrible. x is greater than 11. Approaching these problems like a normal equation is a small, subtle strategy that might not necessarily have any objective “benefit” but definitely helps me psychologically when solving for variables.

**However,** there is one big difference between solving inequalities and solving equations: *If* you multiple or divide both sides by a **negative** number, you must **flip the sign!**

To show you how this works, let me solve the above question again, but I will solve it a bit differently so that we get some negatives.

8x – 20 > 6x + 2 (Distributive property)

– 20 > -2x + 2 (Subtract 8x from both sides)

-22 > -2x (Subtract 2 from both sides)

11 < x (Divide both sides by -2 AND flip the sign around)

You’ll notice that once again we get x is greater than 11 (which we should get again, since we’re solving the exact same equation). This is the one big difference between solving inequalities and solving equations.

As you can imagine, students forget to do this (or do it wrong) quite often when solving inequalities, so I generally encourage students to avoid multiplying or dividing by negatives if possible.

Something people get mixed up about, however, comes with questions like the one below.

4x > -8

Ok, so to get x by itself, we’re going to divide both sides by 4. Now the question is, do we need to flip the sign here?

NO!!!

Even though the number that *gets divided* is negative, the sign stays the same because *we are still dividing BY a positive number).*

So the answer to the above question is x > -2. Notice we keep the sign in the original direction, since we didn’t divide BY a negative.

To conclude, inequalities cause many students to make many mistakes, but these mistakes are very avoidable if you just **solve the equations normally** and **flip the sign if you multiply of divide by a negative number.**

There was an actual SAT question with inequalities I wanted to show, but it seems that will need a blog post of its own.

Till next time!

]]>Today, I want to go over a **big exception **to this rule: questions where “he or she” is the correct answer. I don’t mean the answer is either “he” or “she.” I mean the correct answer actually reads “he or she.”

In my own experience, this might be the *only* time where the answer that sounds right to me (usually “they”) is actually NOT the correct answer, and therefore this is a good rule to know.

Let me give you an example of a sentence in which “he or she” is appropriate.

If a student wants to become a doctor, ** he or she** must take take several biology classes.

A big concept I teach with pronoun questions is finding the noun or nouns the pronoun refers to. If the noun is singular, you need a singular pronoun (like it) while plural nouns require plural pronouns (like they).

Look at the sentence above. What noun(s) does *he or she* refer to? A student, right? Not students. Just a single, solitary student. Therefore, we need a singular pronoun to refer to that student. Now obviously we don’t use the “it” while referencing people, and because we don’t know the gender of the student, we can’t just pick one.

Therefore, the rule is, when you are referring to **one person and you don’t know the gender of that person, ****you use he or she.**

One more related example.

If a student wants to become an architect, ** he or she** must take

Same principle with possessive pronouns. One person, don’t know the gender, use his or her.

Now, if you’re like me, you would probably say “they” or “their” in normal conversation, and this would likely sound right to you when reading the above sentences out loud. However, “they” or “their” would NOT be correct in these instances, and this is why I consider this a big and good-to-know exception to the “sounding it out strategy.”

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Expected value questions often appear towards the end of the math section (questions 50-60) on the ACT and are usually accompanied by a table similar to the one below.

Score on Test | Probability |

60 | .10 |

70 | .15 |

80 | .30 |

90 | .20 |

100 | .25 |

The question usually asks something like “what is the *expected* score this student will get on this test.” These questions do use that specific word *expected* almost every time, which makes them easy to recognize!

First of all, here’s how most people try to solve this question. They add 60+70+80+90+100, and then divide by 5, as if they were finding a normal average. **However**, this doesn’t take into account that certain scores are more likely than other scores. For example, a student has a better chance of getting a 100 than getting a 60, but simply taking the averages of the scores doesn’t take this into account at all.

The method for solving expected value questions is unfamiliar to many but easy enough to learn. What you want to do is multiply each possible score by the probability that it happens, and then add up those products. Sounds harder than it is, so let me show you for this one.

60 x .10 = **6**

70 x .15 = **10.5**

80 x .30 = **24**

90 x .20 = **18**

100 x .25 = **25**

And then we add those products: 6 + 10.5 + 24 + 18 + 25 = **83.5**

And that’s it! 83.5 is the student’s expected score on this test!

Sometimes the chart will give fractions for the probabilities instead of decimals, but *the process is exactly the same!* Multiply each score by the fraction that represents its probability, and then add those up.

This is a great bit of math to know for the ACT, because these questions are usually featured in the “hard question” range, but are really among the easiest to set up and solve! On the SAT, a similar type of question appears in the form of **histograms**, but we’ll cover those another time.

Next time, I will show you examples of how calculating expected value actually has several applications to everyday situations, and hopefully convince a few more people that not all of this math stuff is totally irrelevant to their lives!

Till then!

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For a few types of questions on the SAT Writing (or ACT English) exam, it is not unusual to have two answer choices that mean the same thing. Look at the question below, taken from an actual College Board practice test.

Fred Harvey, an English born entrepreneur. He decided to open his own restaurant business to serve rail customers.

(a) no change

(b) entrepreneur:

(c) entrepreneur; he

(d) entrepreneur,

Alright, clearly a punctuation question. Now before you do anything else, look at the way that it is currently written, and then look at answer choice C. What’s the difference? Well, as written the sentence contains a period, but in C the period has been replaced by a semicolon.

As we now know though, periods and semicolons are **the same thing**! Therefore, there is no way that one would be correct and the other would be incorrect. It would be like one answer on the math section being 3/6, and another answer being 4/8. Those fractions are equal, so there’s no way you could have one be right and the other be wrong.

Armed with this knowledge, you can confidently eliminate both answers A and C for the question above, as they mean the same thing as each other. This exact situation has appeared multiple times on the new SAT, so this is an easy to apply and effective strategy for the writing section.

Another type of question where you can get two answer choices that mean the same thing is transition words.

Here is a question from another College Board test.

Therefore, between 1992 and 1996 more than 400 independent philosophy programs were eliminated from institutions.

(a) no change

(b) Thus,

(c) Moreover,

(d) However,

Look at answers A and B. Therefore and thus. Both of these words are used in the exact same way: to introduce cause and effect. Therefore(thus), we can eliminate both answer choices. Same thing with “furthermore” and “moreover.” Same thing with “however” and “nevertheless.” All of these pairs have appeared together on the new SAT, and knowing which transition words have identical meanings can really help for these questions.

A third type of question where you sometimes see identical answer choices is for wordiness questions. For example, there was a question on an SAT where the four options were

(a) per year

(b) every year

(c) each year

(d) Delete the underlined phrase

What’s the difference between A, B, and C? Those 3 phrases all mean the same thing to me, and I can’t imagine a grammatical rule that makes one of those right and the other two wrong. Good bet we want to delete the underlined phrase.

Now, you don’t want to overuse this strategy. Oftentimes different answer choices will be *similar*, but that doesn’t mean they are the grammatically *identical*. The specific question types in which I have encountered truly *identical* answer choices are

- semicolons and periods
- transition words (thus and therefore, etc)
- wordiness

When using this strategy, always make sure that there isn’t another difference in the answer choice. For example, if one answer choice has a semicolon in place of a period AND an additional word added, then those answers wouldn’t be identical.

Overall though, this is a great strategy for the SAT Writing and ACT English sections, so be on the lookout for these types of questions!

Till next time!

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Well fear not. The rules for semicolons are easy to understand and the strategy for attacking these questions is very straightforward.

Before we get into semicolons specifically I just want to briefly discuss why I devote the first lesson with students to the writing section (or English for the ACT). This section, which covers grammar and english usage, is *very *learnable, and within a coupe of hours you can cover virtually every type of question asked on the section and the specific strategies for answering these questions. *Everybody* can do well on writing, so it is a good way to get momentum going for a test prep course.

Anyways, all you need to know about semicolons for the SAT is **they are the same thing as periods.** They separate independent clauses (things that could be sentences on their own) and the strategy for determining whether to use a semicolon is therefore very straightforward:

1. Look at the phrase before the semicolon. Could that be a sentence by itself?

2. Look at the phrase after the semicolon. Could that be a sentence by itself?

If the answer to **both** of these questions is yes, then a semicolon is appropriate.

Below is an easy example of a semicolon in action.

Tom was very nervous about the performance; his father reassured him and gave him the confidence he needed.

- Could the phrase “Tom was very nervous about his performance” be a sentence by itself? Yes
- Could the phrase “His father reassured him and gave him the confidence he needed” be a sentence by itself? Yes

Two independent clauses, so a semicolon works.

Easy enough, right? Now, let’s change the sentence slightly and see if a semicolon still works.

Tom was very nervous about the performance; **but** his father reassured him and gave him the confidence he needed.

1. Could the phrase “Tom was very nervous about his performance” be a sentence? Still yes.

2. Could the phrase “**But** his father reassured him and gave him the confidence he needed” be a sentence by itself? **NO.**

Now, that second phrase can’t stand alone as a sentence, so we *can’t* use a semicolon (we would use a comma instead).

A few important points here, that tie into strategy for the writing/english section as a whole.

- The strategies for writing are easy to apply, but you must be
**patient and deliberate**in using them. Many of my students know this rule, but don’t take the time to read the entire phrase. This is true of many other types of grammar questions. The good news is that if you take your time and stay disciplined with these strategies you’ll be golden. - Many of these tests in the writing section are easier to apply if you imagine
**saying them out loud.**Things that are wrong in English tend to*sound*wrong, so rather than simply reading sentences it is often helpful to say them in your head. When I was in high school this was really the*only*strategy I employed on writing, and it didn’t fail me. Granted, some people have a harder time doing this, and that is why we also learn these question-specific strategies, but I am a big believer in the ‘what sounds right is probably right’ test. There are one or two specific instances where this intuition might fail you, and I will likely do a blog post about those at some point. (I guess I should also note that some test prep tutors disagree with me on the value of this rudimentary strategy; they’re welcome to write their own blog post).

Tldr: **semicolons = periods.** There are some tricks that you can use to your advantage from the fact that these two things are identical (which comes up with transition words as well) but I think I’m going to have to save that discussion for another day.

Till next time!

]]>Today, we talk more about the actual math concepts related to this problem, and how we can apply these to get the correct answer.

So alright, the question is asking about arc length. Most of you saw questions like this towards the end of the year in geometry, so rack your brains and try to remember what your teacher was going over while you were daydreaming about summer vacation. There’s a proportion that relates arc lengths, the circumference of the circle, and the angle making that arc.

Arc length angle

——————— = ———————

Circumference 360

Look familiar? Alright, well maybe not, but this is a helpful proportion to know for the ACT and SAT (there is a similar equation relating area of a sector in a circle to that circle’s area). Additionally, this should make some intuitive sense. After all, an arc is really just a fraction of the circle’s circumference. And the angle making that arc is just a fraction of the 360 degrees found in every circle. Makes sense that those fractions would equal each other.

Almost there. We know the question is asking us for arc length, and we have an equation that can be used to find arc length, so let’s **plug in what we know.**

The circumference of a circle is 2𝞹r (NOT 𝞹r^2 as countless students have told me in the past) so for this circle we’ve got 2𝞹(4) or 8𝞹.

Arc length angle

———————- = ———————-

8𝞹 360

We can get arc length by itself by multiplying both sides by 8𝞹 (or by cross-multiplying and dividing by 360), and in either case we get

Arc length = 8𝞹(angle)

———————–

360

But what is the angle, you might be asking? Well, that’s what we figured out last time! Because we know the opposite side of the angle is 1 and the hypotenuse is 4, the sine of our angle is (¼) and the angle can then be written as sin-1(¼).

8𝞹(sin-1(¼))

Arc length = —————-

360

Looking very close now. Maybe we can reduce this fraction? 8𝞹 and 360 are both divisible by 8, so we can reduce this we get

1𝞹(sin-1(¼))

Arc length = ——————

45

Or answer choice A.

Phew. Well if nothing else, I hopefully have at least convinced you that this question might just be the hardest the ACT has to offer. As I said last time, however, there are some very important lessons that can be taken from approaching a question like this, so let’s briefly recap those.

- If you find yourself totally mystified by a question on the ACT (particularly in the math section) ask yourself
**what***type***of math concept is this testing?** - A good source of information/inspiration when thinking about a question is
**the answer choices!**We saw sine, cosine, and tangent in all the choices, which gave us a*major*clue about how to narrow down our choices. - If you can think of any equations or helpful rules related to question,
**write them down!**Here we utilized SOH CAH TOA and the arc length proportions, and having them written down in front me me is*extremely*helpful when I’m trying to figure out what to do.

A piece of encouragement to end on: the strategies, techniques, and concepts that we used to solve this problem are *absolutely learnable*, as is everything else on the ACT and SAT. Plus, after this question, everything else will be a walk in the park!

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Not just any problem, however. Over several years of teaching test prep I have worked through each of the official “Bluebook” SATs and “Redbook” ACTs countless times, and the problem below is what always pops into my mind when thinking about the hardest questions I’ve come across.

This question not only looks very difficult at first glance, but requires knowledge and application of some not-so-easy math concepts. However, I think that working through this problem will demonstrate some important test-taking techniques that are applicable throughout the math section, as well as some valuable strategies for attacking questions* even when you don’t really understand it*.

First, without further ado, here is the question we will be looking at:

Looks pretty menacing, huh? The diagram looks tricky and the answer choices, well, they don’t exactly seem user-friendly.

Well OK, first things first, **what information did they give us in the question and what are they asking us to figure out**? We’ve got a right triangle inside of a circle, and we know the lengths of two of the three sides of that triangle. They are asking us for the length of arc AC, which is the section of the circle that the triangle is touching. Now, adding to the confusion is the fact that the triangle touches the circle at point C but NOT at point A (if you look closely, there’s that bit of distance between A and B there).

When I encounter questions like this, where I feel like I have no idea what to do, one important question I ask myself is **“What type of math concept does this seem like?”** Is this Pythagorean Theorem, Angles in a Triangle, Circumference of a Circle, etc. I know that by this point in my math career (and this will almost always be true for juniors and seniors taking the ACT) I have learned something in one of my math classes that will help me solve this problem. If I can identify what general body of math knowledge this problem is related to, then at least I have something specific I can think about.

One thing I will do is **look at the answer choices!** In this problem, they all have sin, cos, and tan, so now I ask myself “What do I remember about sine, cosine, and tangent?” Hopefully, somewhere in the dark recesses of your mind, you come up with:

SOH CAH TOA!

Good bet this is relevant here. All of the answers have 1/4, which I’m guessing has something to do with the two lengths of the triangle they gave me. Alright, well if I can just figure out which combination of opposite, adjacent, and hypotenuse I need, I should at least be able to get my options down to 2 of the multiple choice options.

So 4 is definitely the hypotenuse, which means tangent is out. Now I need to figure out whether side BC (the one with the length of 1) is the opposite side (in which case we need sine) or the adjacent side (cosine). To determine this I need to decide whether this problem involves angle D in the middle (which would make 1 the “opposite” side) or are we talking about angle C near the top (which would make 1 the “adjacent” side)?

Now, remember the question: they’re asking about arc length of a circle. While you might be pretty rusty with your geometry, try to think back to the fourth quarter of geometry class when you were learning about arc lengths and areas of sectors. Which seems more relevant to those problems: the size of the central angle of the circle, or the size of the “I don’t even think there’s a term for it because we never use it” random angle at the top of the triangle that’s connected to the outside of the triangle.

If you guessed central angle D, you’re right!

Now, at last, we have at least enough to narrow down our answer choices. Relative to the central angle, we know that the opposite side (O) is 1, and the hypotenuse is 4 (H). Therefore, even without really knowing how to solve any of this, we should be able to narrow our two options down to F and H, as those are the only two that use sine, which SOH CAH TOA tells us is the one that uses the **O**pposite side and the **H**ypotenuse.

It looks like this problem will require a second post to complete, so join us next time, when we find out whether (F) or (H) is the answer we’re looking for, and review the strategies and test-taking tricks that helped us get there (and will help us with other head-scratching math questions).

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First and foremost, students weighing which test is best for them should **take a full-length practice test for each!** You want these diagnostic tests to simulate testing conditions as much as possible, so a few important guidelines are:

- Take them
**timed!**As we discussed last time, there are differences in pacing and timing between the two tests, which factors heavily into how students score and how students feel moving through each section. - Take them in a
**quiet, distraction free environment.**The “which day of the week are you” quiz can wait until you’re done. - Make sure you are
**awake and fully functional**on the days you take them. These tests do eat up a lot of time, so a good goal might be doing one test on a Saturday one weekend and then the other test the following Saturday.

After students take a practice ACT and a practice SAT, the biggest thing I look at is their **overall percentile on each**. Generally, whichever test a student scores higher on in their initial diagnostic is the one I recommend focusing on most heavily.

There are, I suppose, a few exceptions to this, and I try to take the following factors into account with my students when developing a plan for test-prepping.

**Personal preference/confidence:**Sometimes I have students who felt a lot more comfortable taking the test they initially scored worse on, and feel like with practice they can do better on that one. This is most commonly the case when students feel like they’ve forgotten a lot of the relevant math concepts tested, but with practice they could master those concepts more easily. For example, I’ve had students score higher on the ACT initially, but feel a lot more confident about the algebraic concepts emphasized on the SAT and want to focus on improving there.**Logistical:**Another big consideration (especially for seniors trying to prepare for these tests) can be logistical: the ACT and SAT offer different administrations throughout the year, and occasionally there are situations where a student may not have quite enough time to prepare for the October SAT, and have college applications due before the next available date. In these cases it is important to look carefully at what the students’ various deadlines are, how much time they have before different test dates, and get a realistic sense of how much time the students will have to prepare before each test date. Overall, however, these situations are avoidable, and speak to the importance of planning out test-prep**far in advance**.

I think the discussion of ideal timelines for test-prep is important enough to merit its own blog post (we’ll probably want to get into that next time), but I do want to end with a couple thoughts and observations I’ve had pertaining to today’s topic of deciding between the ACT and the SAT.

- Math counts for 25% of the score on the ACT and 50% of the score on the SAT, so (theoretically at least) students whose strong suit is math would have better chances on the SAT, and students who struggle in math might prefer the ACT.
- The hardest sections to finish on-time tend to be ACT science, ACT reading, and SAT math no-calculator. Therefore, slower readers might struggle more on the ACT, and those who need more time for math problems and feel more pressured under time might struggle more on SAT math (however, people do sometimes have a hard time finishing ACT math as well, which is why taking a diagnostic is so important)
- I have had many students start off lower on the SAT and then do better on it when they take the real one (and vice-versa) so it is very often a good idea to prepare for both and take each at least once. These tests remain an important part of the college application process, and getting the best possible score you can is definitely worth the time and effort it takes to prepare for them!

Next time, I will want to talk about developing a timeline for standardized test prep and perhaps build on that last point about taking both tests vs. focusing your energy on just one. Till then!

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