
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 $E: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) passing through point $P(3, 1)$, with its left and right foci being $F_1$ and $F_2$ respectively, and $\overrightarrow{F_{1}P} \cdot \overrightarrow{F_{2}P} = -6$.

(1) Find the equation of ellipse $E$;  
(2) If $M$ and $N$ are two moving points on the line $x=5$, and $F_1M \perp F_2N$, then does the circle $C$ with diameter $MN$ pass through a fixed point? Please explain your reasoning.

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