
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
Given a positive integer \( n \), let \( P(x) \) and \( Q(x) \) be polynomials with real coefficients of degree no greater than \( n \). They satisfy the equation \( x^{n+1} P(x) + (x+1)^{n+1} Q(x) = 1 \). Find \( Q(x) \), and determine the value of \( Q\left(-\frac{1}{2}\right) \).

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