
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 $C$: $\frac{{x}^{2}}{{a}^{2}}+\frac{{y}^{2}}{{b}^{2}}=1\left(a \gt b \gt 0\right)$ with left and right foci $F_{1}$ and $F_{2}$, and eccentricity $\frac{1}{3}$, a circle with the origin as its center and the length of the minor axis of the ellipse as its radius is tangent to the line $x-y-4=0$. <br/>$(1)$ Find the standard equation of the ellipse $C$; <br/>$(2)$ A line $l$ passing through point $F_{2}$ intersects the ellipse $C$ at points $M$ and $N$ (line $l$ does not coincide with the $x$-axis). Does there exist a point $P$ on the $x$-axis such that the product of the slopes of lines $PM$ and $PN$ is a constant? If so, find the coordinates of all points $P$ that satisfy this condition; if not, explain why.

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