
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 $\overrightarrow{a}=(\cos \omega x-\sin \omega x,\sin \omega x)$ and $\overrightarrow{b}=(-\cos \omega x-\sin \omega x,2\sqrt{3}\cos \omega x)$, let the function $f(x)=\overrightarrow{a}\cdot \overrightarrow{b}+\lambda$ $(x\in\mathbb{R})$ have its graph symmetric about the line $x=\pi$, where $\omega$ and $\lambda$ are constants, and $\omega\in(\frac{1}{2} ,1)$.

$(1)$ Find the smallest positive period of the function $f(x)$;

$(2)$ If the graph of $y=f(x)$ passes through the point $(\frac{\pi }{4} ,0)$, find the range of values of the function $f(x)$ in the interval $\left[0,\frac{3\pi }{5} \right]$.

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