
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 $f(x)=2\sin x \cos x - 2\sqrt{3}\sin^2 x + \sqrt{3}$.
$(1)$ Simplify the analytical expression of the function $f(x)$;
$(2)$ Find the range of the function $f(x)$ on the interval $[0, \frac{\pi}{2}]$;
$(3)$ Let $α \in (\frac{\pi}{2}, \pi)$, if $f(\frac{α}{2}) = \frac{10}{13}$, find the value of $\sin \alpha$.

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