
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 short axis endpoint and the two foci forms a square, and the equation of the inscribed circle 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 of the foci $F$ of the ellipse $C$, the line $l: y = x + m$ intersects the parabola $E$ at two points $A, B$, and $0 \leqslant m \leqslant 1$, find the maximum value of the area of $\triangle FAB$.

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