
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 $\overrightarrow{a}=({2\sin x, 2\cos x})$ and $\overrightarrow{b}=({3\sin x+4\cos x, -\cos x})$, let $f(x)=\overrightarrow{a}\cdot\overrightarrow{b}$.  
$(1)$ Find the maximum value of the function $f(x)$.  
$(2)$ Given that in acute triangle $\triangle ABC$, where the sides opposite angles $A$, $B$, and $C$ are $a$, $b$, and $c$ respectively, and it satisfies $f\left(\frac{B}{2}+\frac{\pi}{4}\right)=\frac{4c}{a}+2$, find the range of $\sin B \cdot \sin C$.

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