
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
Expand the function \( y = \alpha + \beta \cos \vartheta + \gamma \cos^{2} \vartheta \) into a Fourier series on the interval \( (0, \pi) \) with respect to the system \( P_{0}(\cos \vartheta), P_{1}(\cos \vartheta), P_{2}(\cos \vartheta), \ldots \), where \( P_{n} \) are Legendre polynomials.

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