
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
Consider the identity \( 1+2+\cdots+n=\frac{1}{2} n(n+1) \). If we set \( P_{1}(x)=\frac{1}{2} x(x+1) \), then it is the unique polynomial such that for all positive integer \( n \), \( P_{1}(n)=1+2+\cdots+n \). In general, for each positive integer \( k \), there is a unique polynomial \( P_{k}(x) \) such that
\[ P_{k}(n)=1^{k}+2^{k}+3^{k}+\cdots+n^{k} \quad \text { for each } n=1,2, \ldots . \]

Find the value of \( P_{2010}\left(-\frac{1}{2}\right) \).

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