
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 left focus $F$ and a chord perpendicular to the major axis of length $6\sqrt{2}$, a line passing through point $P(2,1)$ with slope $-1$ intersects $C$ at points $A$ and $B$, where $P$ is the midpoint of $AB$. Find the maximum distance from a point $M$ on ellipse $C$ to focus $F$.

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