
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
Let $C\_1$ be a parabola with focus $F$ and equation $y^{2}=2px (p > 0)$, and let $C\_2$ be a hyperbola with asymptotes $2x-\\sqrt{3}y=0$ and $2x+\\sqrt{3}y=0$, and one focus at $(0, \\sqrt{7})$.

(1) Find the standard equation of the hyperbola $C\_2$.

(2) If $C\_1$ and $C\_2$ have two common points $A$ and $B$ in the first quadrant, find the range of $p$ and the maximum value of $\\overrightarrow{FA}\\cdot \\overrightarrow{FB}$.

(3) Under the conditions of (2), if the area $S$ of $\\triangle FAB$ satisfies $S=\\dfrac{2}{3}\\overrightarrow{FA}\\cdot \\overrightarrow{FB}$, find the value of $p$.

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