
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 $\overrightarrow{a}=(\sin x, \sin x)$ and $\overrightarrow{b}=(\sqrt{3}\cos x, \sin x)$,
(Ⅰ) Define the function $f(x)=\overrightarrow{a}\cdot \overrightarrow{b}$. Find the intervals where $f(x)$ is monotonically increasing.
(Ⅱ) In triangle $ABC$, acute angle $A$ satisfies $f(A)=\frac{3}{2}$, $b+c=4$, and $a=\sqrt{7}$. Find the area of triangle $ABC$.

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