
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 vectors $\overset{⇀}{m}=(\cos x,-\sin x)$ and $\overset{⇀}{n}=(\cos x,\sin x-2 \sqrt{3}\cos x)$, where $x\in\mathbb{R}$, let $f(x)= \overset{⇀}{m}\cdot \overset{⇀}{n} = \frac{24}{13}$, and $x\in\left[ \frac{\pi}{4}, \frac{\pi}{2}\right]$, then the value of $\sin 2x$ is __________.

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