
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 $\vec{a}=(\cos x,\sin x)$, $\vec{b}=(-\cos x,\cos x)$, $\vec{c}=(-1,0)$  
$(1)$ If $x= \frac{\pi}{6}$, find the angle between vectors $\vec{a}$ and $\vec{c}$;  
$(2)$ When $x\in\left[ \frac{\pi}{2}, \frac{9\pi}{8}\right]$, find the minimum value of the function $f(x)=2\vec{a}\cdot \vec{b}+1$.

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