
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.** Anka wants to know if it hurt when her crush fell from the sky. She curls up into a ball and jumps off a  $80 $  meter high building. If she bounces up to  $3/4$  of the previous height each bounce, how many times can she bounce while still moving a total distance of less than  $300$  meters?**p2.** Alex wants to rearrange the alphabet to put him and his crush next to each other. If he randomly rearranges it, what is the probability that “u” and “i” are next to each other?**p3.** Jeffrey, being from Tennessee, sees  $10$ s everywhere he looks. If he assigns to each of his 1 $0000$  lovers a unique integer ID number from  $1$  to  $10000$  how many of them will have the sequence “ $10$ ” in their ID?**p4.** Andrew is getting lost in Amy’s eyes, or more specifically, her  $i$ ’s:  $a_i , b_i , c_i , ..., z_i $ . 
Let  $a_n =\frac{1}{2^n}$  , $b_n =\frac{1}{3^n}$ ,  $...$ ,  $z_n =\frac{1}{27^n}$  .
Additionally, let  $S = \{(i_1,i_2,...,i_{26}) | i_1 \ge 1,i_2,i_3,...,i_{26} \ge 0,i_1,i_2,...,i_{26} \in Z\}$ .
Find the sum of  $a_{i_1}b_{i_2}... z_{i_{26}}$  over all  $(i_1,i_2,...,i_{26}) \in S$ .**p5.** Janabel is in love with regular pentagon  $ANGEL$  with side length  $4$  and area  $x$ . Find  $x^2$ .


PS. You should use hide for answers.

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