
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
**p22.** Consider the series  $\{A_n\}^{\infty}_{n=0}$ , where  $A_0 = 1$  and for every  $n > 0$ ,   $$ A_n = A_{\left[ \frac{n}{2023}\right]} + A_{\left[ \frac{n}{2023^2}\right]}+A_{\left[ \frac{n}{2023^3}\right]}, $$    where  $[x]$  denotes the largest integer value smaller than or equal to  $x$ . Find the  $(2023^{3^2}+20)$ -th element of the series.**p23.** The side lengths of triangle  $\vartriangle ABC$  are  $5$ ,  $7$  and  $8$ . Construct equilateral triangles  $\vartriangle A_1BC$ ,  $\vartriangle B_1CA$ , and  $\vartriangle C_1AB$  such that  $A_1$ , $B_1$ , $C_1$  lie outside of  $\vartriangle ABC$ . Let  $A_2$ , $B_2$ , and  $C_2$  be the centers of  $\vartriangle A_1BC$ ,  $\vartriangle B_1CA$ , and  $\vartriangle C_1AB$ , respectively. What is the area of  $\vartriangle A_2B_2C_2$ ?**p24.**There are  $20$  people participating in a random tag game around an  $20$ -gon. Whenever two people end up at the same vertex, if one of them is a tagger then the other also becomes a tagger. A round consists of everyone moving to a random vertex on the  $20$ -gon (no matter where they were at the beginning). If there are currently  $10$  taggers, let  $E$  be the expected number of untagged people at the end of the next round. If  $E$  can be written as  $\frac{a}{b}$  for  $a, b$  relatively prime positive integers, compute  $a + b$ .


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

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