
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 $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1 (a > b > 0)$ with left and right foci $F\_1$ and $F\_2$, respectively. Point $A(4,2\sqrt{2})$ lies on the ellipse, and $AF\_2$ is perpendicular to the $x$-axis.

1. Find the equation of the ellipse.
2. A line passing through point $F\_2$ intersects the ellipse at points $B$ and $C$. Find the maximum area of triangle $COB$.

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