
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 the ellipse $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$, the line connecting the endpoint of the minor axis and the two foci of the ellipse forms a square. The equation of the inscribed circle of this square is $x^{2}+y^{2}=2$.
(1) Find the equation of the ellipse $C$;
(2) If the focus of the parabola $E$: $y^{2}=2px(p > 0)$ coincides with one focus of the ellipse $C$, and the line $l$: $y=x+m$ intersects the parabola $E$ at two points $A$ and $B$, with $0\leqslant m\leqslant 1$, find the maximum value of the area of $\triangle FAB$.

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