
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 vector $\overrightarrow{a}=\left( \frac {1}{2}, \frac {1}{2}\sin x+ \frac { \sqrt {3}}{2}\cos x\right)$ is collinear with $\overrightarrow{b}=(1,y)$, let $y=f(x)$.
$(1)$ Find the period and the maximum value of the function $f(x)$;
$(2)$ In an acute triangle $ABC$ with internal angles $A$, $B$, $C$, if $f(A- \frac {\pi}{3})= \sqrt {3}$ and $BC= \sqrt {7}$, $\sin B= \frac { \sqrt {21}}{7}$, find the area of triangle $ABC$.

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