
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 vector $\overrightarrow{a}=(\sin x,\cos x)$, $\overrightarrow{b}=(\cos x,\cos x)$, where $x\in\mathbb{R}$, and the function $f(x)= \overrightarrow{a}\cdot( \overrightarrow{a}+ \overrightarrow{b})$.

$(1)$ Find the maximum value of the function $f(x)$;  
$(2)$ Find the intervals of monotonic increase for the function $f(x)$ on $[0,\pi]$.

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