
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 $\overrightarrow{m}=(f(x),2\cos x)$ and $\overrightarrow{n}=(\sin x+\cos x,1)$, and $\overrightarrow{m}$ is parallel to $\overrightarrow{n}$.  
$(1)$ Find the expression for the function $f(x)$.  
$(2)$ If the graph of the function $f(x)$ is moved downward by $1$ unit, and then, keeping the y-coordinate unchanged, the x-coordinate is reduced to half of its original value, resulting in the graph of function $g(x)$. Find the maximum value of the function $g(x)$ on $x\in[0, \frac{\pi}{8}]$ and the corresponding $x$ value.

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