
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
Decompose the function \( y=(1-x^{2}) x^{\nu}(\nu>-1) \) into a Fourier series on the interval \( (0,1) \) using the system of functions \( J_{\nu}\left(\mu_{n} x\right), n=1,2, \ldots \), where \( J_{\nu} \) is the Bessel function and \( \mu_{n} \) is the zero of the function \( J_{\nu} \).

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