
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
Let  $P(x) = x^2-1$  be a polynomial, and let  $a$  be a positive real number satisfying  $$ P(P(P(a))) = 99. $$   The value of  $a^2$  can be written as  $m+\sqrt{n}$ , where  $m$  and  $n$  are positive integers, and  $n$  is not divisible by the square of any prime. Find  $m+n$ .

*Proposed by **HrishiP***

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