
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
**p1.**  $4$  balls are distributed uniformly at random among  $6$  bins. What is the expected number of empty bins?**p2.** Compute  ${150 \choose 20 }$  (mod  $221$ ).**p3.** On the right triangle  $ABC$ , with right angle at $ B$ , the altitude  $BD$  is drawn.  $E$  is drawn on  $BC$  such that AE bisects angle  $BAC$  and F is drawn on  $AC$  such that  $BF$  bisects angle  $CBD$ . Let the intersection of  $AE$  and  $BF$  be  $G$ . Given that  $AB = 15$ , $ BC = 20$ ,  $AC = 25$ , find  $\frac{BG}{GF}$  .**p4.** What is the largest integer  $n$  so that  $\frac{n^2-2012}{n+7}$  is also an integer?**p5.** What is the side length of the largest equilateral triangle that can be inscribed in a regular pentagon with side length  $1$ ?**p6.** Inside a LilacBall, you can find one of  $7$  different notes, each equally likely. Delcatty must collect all  $7$  notes in order to restore harmony and save Kanto from eternal darkness. What is the expected number of LilacBalls she must open in order to do so?


PS. You had better use hide for answers.

Present the answer in LaTex format: \boxed{Your answer}
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