
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
An ellipse $C: \frac{x^2}{a^2} + \frac{y^2}{b^2} = 1$ ($a > b > 0$) has its upper vertex at $A(0,b)$, and a point $P(\frac{4}{3},\frac{b}{3})$ lies on the ellipse $C$. A circle with diameter $AP$ passes through the right focus point $F$ of the ellipse $C$.

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

$(2)$ Suppose a moving line $l$ intersects ellipse $C$ at only one point, and there exist two fixed points on the x-axis such that the product of their distances to line $l$ is $1$. Determine the coordinates of these two fixed points.

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