
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 $\overrightarrow{m}=( \sqrt {3}\sin x,\cos x)$, $\overrightarrow{n}=(\cos x,\cos x)$, $(x\in \mathbb{R})$, let $f(x)= \overrightarrow{m}\cdot \overrightarrow{n}$.
$(1)$ Find the analytic expression for $f(x)$ and its interval of monotonic increase;
$(2)$ In $\triangle ABC$, let $a$, $b$, $c$ be the sides opposite to angles $A$, $B$, $C$ respectively, and given $a=1$, $b+c=2.f(A)=1$, find the area of $\triangle ABC$.

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