
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 $\vec{a}=(\sin (x+ \frac{\pi}{3}),\sin (x- \frac{\pi}{6}))$, $\vec{b}=(\cos (x- \frac{\pi}{6}),\cos (x+ \frac{\pi}{3}))$, and $\vec{a} \cdot \vec{b}= \frac{5}{13}$, with $x\in[- \frac{\pi}{3}, \frac{\pi}{6}]$, calculate the value of $\sin 2x$.

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