
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 three points $P(5,2)$, ${F_1}(-6,0)$, and ${F_2}(6,0)$.

(1) Find the standard equation of the ellipse with foci at ${F_1}$ and ${F_2}$ and passes through point $P$.

(2) Point $M$ is a point on the ellipse. If $MF_{1}=2\sqrt{5}$, find the length of $MF_{2}$.

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