
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 $f(x)=\sqrt{3}\sin\frac{ωx}{2}\cdot\cos\frac{ωx}{2}-\cos^{2}\frac{ωx}{2}$ ($ω>0$) is monotonically increasing on $(0,\frac{4π}{3})$ and monotonically decreasing on $(\frac{4π}{3},2π)$. If the function $f(x)$ is monotonic on $(\frac{17π}{3},a)$, determine the maximum value of $a$.

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