
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 $\overrightarrow{m}=(\sqrt{3}\sin x,\cos x)$ and $\overrightarrow{n}=(\cos x,\cos x)$, where $x\in \mathbb{R}$, let $f(x)=\overrightarrow{m}\cdot\overrightarrow{n}$.  
$(1)$ Find the analytical expression of the function $f(x)$ and the interval where it is monotonically increasing.  
$(2)$ In triangle $ABC$, with $a$, $b$, $c$ being the opposite sides of angles $A$, $B$, $C$ respectively, and $a=1$, $b+c=2$, if $f(A)=1$, find the area of triangle $ABC$.

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