
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 the function $f(x)=x^{3}+\sin x$, if $\alpha\in[0,\pi]$ and $\beta\in\left[-\frac{\pi}{4}, \frac{\pi}{4}\right]$ and $f\left(\frac{\pi}{2}-\alpha\right)=f(2\beta)$, then $\cos\left(\frac{\alpha}{2}+\beta\right)=$ ______.

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