
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 $\overrightarrow{m}=( \sqrt {3}\sin x,\cos x)$, $\overrightarrow{n}=(\cos x,\cos x)$, where $x\in R$, let $f(x)= \overrightarrow{m}\cdot \overrightarrow{n}$.
(I) Find the analytical expression of function $f(x)$ and its interval of monotonic increase;
(II) In $\triangle ABC$, $a$, $b$, $c$ are the sides opposite to angles $A$, $B$, $C$ respectively, with $a=1$, $b+c=2$, and $f(A)=1$. Find the area of $\triangle ABC$.

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