
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 an ellipse $C$: $\frac{x^{2}}{a^{2}}+\frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with left and right foci $F_{1}(-c,0)$ and $F_{2}(c,0)$, respectively. A line $l_{1}$ is drawn through $F_{2}$ perpendicular to the $x$-axis, intersecting the ellipse $C$ at points $A$ and $B$, such that $|AF_{1}|=7|AF_{2}|$.

(1) Find the eccentricity of the ellipse $C$.
(2) A line $l_{2}$ with slope $1$ is drawn through point $F_{1}$, intersecting $C$ at points $M$ and $N$. If the area of $\triangle OMN$ is $\frac{2\sqrt{6}}{5}$, find the equation of the ellipse $C$.

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