
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 $f(x)=a\sin x+b\cos x$ where $ab\neq 0$, and the graph of the function is symmetric about $x=\frac{π}{6}$. If $f({x_0})=\frac{8}{5}a$, then determine the value of $\sin({2{x_0}+\frac{π}{6}})$.

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