
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 $C$: $\frac{x^{2}}{a^{2}}+ \frac{y^{2}}{b^{2}}=1(a > b > 0)$ with foci $F\_1$, $F\_2$, and one endpoint of the minor axis at point $P$. The radius of the incircle of $\triangle PF\_1F\_2$ is $\frac{b}{3}$. Let the line segment $RS$ be the intersection of the line $l$ passing through point $F\_2$ and the ellipse $C$. When $l \perp x$ axis, $|RS|=3$.

(1) Find the standard equation of the ellipse $C$;

(2) Does there exist a point $T$ on the $x$ axis such that, as $l$ varies, $TS$ and $TR$ are always symmetric about the $x$ axis? If so, find the coordinates of point $T$. If not, explain the reason.

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