
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 its major axis being $2\sqrt{2}$ times its minor axis, and the ellipse passes through point $A(2,\frac{\sqrt{2}}{2})$.

(1) Find the equation of the ellipse $(C)$;

(2) Let line $(l)$, which is not parallel to the coordinate axes, intersect the ellipse $(C)$ at points $M$ and $N$ with $|MN|=2\sqrt{2}$. Denote the intercept of line $(l)$ on the $y$-axis as $m$. Find the maximum value of $m$.

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