
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
For each positive integer  $n$  we denote by  $p(n)$  the greatest square less than or equal to  $n$ .
[list=a]
[*]Find all pairs of positive integers  $(  m,n)$ , with  $m\leq n$ , for which
\[ p(  2m+1) \cdot p(  2n+1)  =400 \]
[*]Determine the set  $\mathcal{P}=\{  n\in\mathbb{N}^{\ast}\vert n\leq100\text{ and }\dfrac{p(n+1)}{p(n)}\notin\mathbb{N}^{\ast}\}$ [/list]

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