
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 $\overset{⇀}{a} =(1,-x)$ and $\overset{⇀}{b} =(x^{2},4\cos \theta)$, and the function $f(x)=\overset{⇀}{a} \cdot \overset{⇀}{b} -1$, where $\theta \in [-\pi,\pi]$.

$(1)$ When $\theta =\frac{2\pi }{3}$, find the maximum and minimum values of the function $f(x)$ on the interval $[-2,2]$;

$(2)$ If the function $f(x)$ is not monotonic on the interval $[1,\sqrt{2}]$, find the range of values for $\theta$.

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