
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 $\vec{m}=(2a\cos x,\sin x)$ and $\vec{n}=(\cos x,b\cos x)$, the function $f(x)=\vec{m}\cdot \vec{n}-\frac{\sqrt{3}}{2}$, and $f(x)$ has a y-intercept of $\frac{\sqrt{3}}{2}$, and the closest highest point to the y-axis has coordinates $\left(\frac{\pi}{12},1\right)$.

$(1)$ Find the values of $a$ and $b$;

$(2)$ Move the graph of the function $f(x)$ to the left by $\varphi (\varphi > 0)$ units, and then stretch the x-coordinates of the points on the graph by a factor of $2$ without changing the y-coordinates, to obtain the graph of the function $y=\sin x$. Find the minimum value of $\varphi$.

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