
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
The top vertex of an ellipse $C : \frac{x^2}{a^2}+ \frac{y^2}{b^2} = 1$ $(a > b > 0)$ is $A$, and $P\left( \frac{4}{3}, \frac{b}{3} \right)$ is a point on $C$. A circle with diameter $AP$ passes through the right focus $F$ of the ellipse $C$.

(1) Find the equation of ellipse $C$.

(2) A moving line $l$ intersects ellipse $C$ at exactly one point. Is it possible to find two fixed points on the $x$-axis such that the product of their distances to line $l$ is equal to 1? If yes, find the coordinates of these two fixed points; if no, explain why.

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