
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  $\mathcal{P}$  be the set of all polynomials  $p(x)=x^4+2x^2+mx+n$ , where  $m$  and  $n$  range over the positive reals. There exists a unique  $p(x) \in \mathcal{P}$  such that  $p(x)$  has a real root,  $m$  is minimized, and  $p(1)=99$ . Find  $n$ . 

*Proposed by **AOPS12142015***

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