
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 $\overrightarrow{a}=(\sqrt{2}\sin x,\sqrt{2}\cos x)$ and $\overrightarrow{b}=(\sin x,2\cos x-\sin x)$, let $f(x)=\overrightarrow{a}\cdot\overrightarrow{b}-\frac{3\sqrt{2}}{2}$.  
$(1)$ Find the value of $f(\frac{\pi}{4})$.  
$(2)$ For $x\in(-\frac{\pi}{4},\frac{\pi}{8})$, if $g(x)=\sin 4x+\sqrt{2}mf(x)-3$ has a root, find the range of real number $m$.

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