
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
Let $F_{1}$ and $F_{2}$ be the left and right foci of the ellipse $\frac{{x}^{2}}{25}+\frac{{y}^{2}}{16}=1$, respectively. Let $P$ be any point on the ellipse, and the coordinates of point $M$ are $\left(6,4\right)$. Find the maximum value of $|PM|+|PF_{1}|$.

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