
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 $C$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1\left(a \gt b \gt 0\right)$ passing through the point $P(2,1)$, where $F_{1}$ and $F_{2}$ are the left and right foci of the ellipse $C$, and $|PF_{1}|+|PF_{2}|=4\sqrt{2}$. <br/>$(1)$ Find the standard equation of the ellipse $C$; <br/>$(2)$ Points $M$ and $N$ are two points on the ellipse $C$ that are not coincident with point $P$. A circle with diameter $MN$ passes through point $P$. If the line $MN$ passes through a fixed point, find the coordinates of that point; if it does not pass through a fixed point, explain the reason.

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