
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 the function $f(x)=m\sin x+ \sqrt{2}\cos x$ ($m > 0$) has a maximum value of $2$.

$(1)$ Find the interval of monotonic decrease of $f(x)$ on $[0,\pi]$;

$(2)$ In $\triangle ABC$, $f\left(A- \frac{\pi}{4}\right)+f\left(B- \frac{\pi}{4}\right)=4 \sqrt{6}\sin A\sin B$, the angles $A$, $B$, $C$ are opposite to the sides $a$, $b$, $c$ respectively, and $C=60^{\circ}$, $c=3$, find the area of $\triangle ABC$.

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