
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 an eccentricity of $\frac{\sqrt{2}}{2}$, and the ellipse $C$ passes through point $(\sqrt{3},-\frac{\sqrt{2}}{2})$. Two perpendicular lines ($l_{1}$) and ($l_{2}$) are drawn through point $(1,0)$, intersecting ellipse $C$ at points $P$, $Q$, $M$, and $N$.

(I) Find the standard equation of ellipse $C$.

(II) If $\overrightarrow{MS}=\overrightarrow{SN}$ and $\overrightarrow{PT}=\overrightarrow{TQ}$, investigate whether line $ST$ passes through a fixed point. If it does, find the coordinates of the fixed point; if not, explain the reason.

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