
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 $\overset{→}{a}=(\sin x,\sin x)$, $\overset{→}{b}=( \sqrt{3}\cos x,\sin x)$,

(1) Define the function $f(x)= \overset{→}{a}· \overset{→}{b}$, and find the interval(s) where $f(x)$ is monotonically increasing.

(2) In $\triangle ABC$, the acute angle $A$ satisfies $f(A)= \dfrac{3}{2}, b+c=4, a= \sqrt{7}$. Find the area of $\triangle ABC$.

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