
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 $f(x)=2\sin (2x-\frac{\pi }{4})$

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

$(2)$ If the function $f(x)$ reaches its maximum value at $x=x_0$, find the value of $f(x_0)+f(2x_0)+f(3x_0)$.

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