
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
**p1A.** Compute
           $$ 1 + \frac{1}{2^3} +  \frac{1}{3^3} +  \frac{1}{4^3} +  \frac{1}{5^3} + ... $$              $$ 1 -  \frac{1}{2^3} +  \frac{1}{3^3} -  \frac{1}{4^3} +  \frac{1}{5^3} - ... $$  **p1B.** Real values  $a$  and  $b$  satisfy  $ab = 1$ , and both numbers have decimal expansions which repeat every five digits:
        $$   a = 0.(a_1)(a_2)(a_3)(a_4)(a_5)(a_1)(a_2)(a_3)(a_4)(a_5)... $$            and
         $$  b = 1.(b_1)(b_2)(b_3)(b_4)(b_5)(b_1)(b_2)(b_3)(b_4)(b_5)... $$  If  $a_5 = 1$ , find  $b_5$ .**p2.**  $P(x) = x^4 - 3x^3 + 4x^2 - 9x + 5$ .   $Q(x)$  is a  $3$ rd-degree polynomial  whose graph intersects the graph of  $P(x)$  at  $x = 1$ ,  $2$ ,  $5$ , and  $10$ .  Compute   $Q(0)$ .**p3.** Distinct real values  $x_1$ ,  $x_2$ ,  $x_3$ ,  $x_4 $ all satisfy  $ ||x - 3| - 5| = 1.34953$ .  Find  $x_1 + x_2 + x_3 + x_4$ .**p4.** Triangle  $ABC$  has sides  $AB = 8$ ,   $BC = 10$ , and  $CA = 11$ .  Let  $L$  be the locus of points in the interior of triangle  $ABC$  which are within one unit  of either  $A$ ,  $B$ , or  $C$ .  Find the area of  $L$ .**p5.** Triangles  $ABC$  and  $ADE$  are equilateral, and  $AD$  is an altitude of  $ABC$ .  The area of the intersection of these triangles is  $3$ .  Find the area of  the larger triangle  $ABC$ .


PS. You had better use hide for answers. Collected [here](https://artofproblemsolving.com/community/c5h2760506p24143309).

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