
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  $a_0, a_1,\dots, a_{19} \in \mathbb{R}$  and   $$ P(x) = x^{20} + \sum_{i=0}^{19}a_ix^i, x \in \mathbb{R}. $$   If  $P(x)=P(-x)$  for all  $x \in \mathbb{R}$ , and   $$ P(k)=k^2, $$   for  $k=0, 1, 2, \dots, 9$  then find   $$ \lim_{x\rightarrow 0} \frac{P(x)}{\sin^2x}. $$  

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