
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
Let the function be $f(x)=2\sin x(\sin x+\cos x)$.

$(1)$ Find the smallest positive period and the intervals of monotonic increase for $y=f(x)$;

$(2)$ Solve $f(x)-2=0$ in the interval $\left[ 0,\frac{\pi}{2} \right]$ for two roots $x_1,x_2$, and find the value of $\sin \left( 2x_1+2x_2 \right)$.

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