
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 eccentricity $\frac{\sqrt{2}}{2}$, and the ellipse $C$ passes through the point $(\sqrt{3}, -\frac{\sqrt{2}}{2})$. Two perpendicular lines $l_1$ and $l_2$ are drawn through the point $(1,0)$ and intersect the ellipse $C$ at points $P$, $Q$, $M$, and $N$ respectively.  
(1) Find the standard equation of the ellipse $C$;  
(2) If $\overrightarrow{MS} = \overrightarrow{SN}$, $\overrightarrow{PT} = \overrightarrow{TQ}$, investigate whether the line $ST$ passes through a fixed point. If yes, find the coordinates of the fixed point; if not, explain why.

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