
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 that the center of the ellipse $C$ is at the origin of the coordinates, the focus is on the $x$-axis, one of its vertices coincides with the focus of the parabola $y=\frac{1}{4}{{x}^{2}}$, and the eccentricity is $\frac{2\sqrt{5}}{5}$.

(1) Find the standard equation of the ellipse $C$.

(2) A line $l$ passes through the right focus $F$ of the ellipse $C$ and intersects the ellipse $C$ at points $A$ and $B$, and intersects the $y$-axis at point $M$. If $\overrightarrow{MA}=m\overrightarrow{FA}$ and $\overrightarrow{MB}=n\overrightarrow{FB}$, find the value of $m+n$.

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