
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 $\overrightarrow{a} =(-\sin x,2)$ and $\overrightarrow{b} =(1,\cos x)$, and the function $f(x)=\overrightarrow{a} \cdot\overrightarrow{b}$

(1) Find the value of $f\left( \frac{\pi}{6} \right)$

(2) If $\overrightarrow{a} \perp \overrightarrow{b}$, find the value of $g(x)=\frac{\sin \left(π+x\right)+4\cos (2π-x)}{\sin \left( \frac{π}{2}-x\right)-4\sin (-x)}$

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