
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 function $f(x) = \begin{cases}
x^2 + \sin\left(x + \frac{\pi}{3}\right), & \text{if } x > 0 \\
-x^2 + \cos\left(x + \alpha \right), & \text{if } x < 0
\end{cases}$, where $\alpha \in [0, 2\pi)$, and $f$ is an odd function, then $\alpha = \_\_\_\_\_\_$.

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