
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 $x^3+\sin 2x = m$, $y^3+\sin 2y = -m$, and $x, y \in \left(-\frac{\pi}{4}, \frac{\pi}{4}\right)$, $m \in \mathbb{R}$, then $\tan\left(x+y+\frac{\pi}{3}\right) = \_\_\_\_\_\_$.

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