
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 $C:x^{2}+\frac{y^{2}}{m}=1$ passes through point $M(\frac{\sqrt{2}}{2},1)$.

(I) Find the equation of ellipse $C$, the coordinates of its foci, and its eccentricity;

(II) Suppose the foci of ellipse $C$ are $F_{1}$ and $F_{2}$. The line $l:y=kx+1 (k\neq 0)$ passing through focus $F_{2}$ intersects $C$ at points $A$ and $B$. When line $MF_{2}$ bisects $\angle AMB$, find the area of $\triangle AB{F_{1}}$.

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