
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 $\frac{\sqrt{2}}{2}$, its left and right foci are $F_1$ and $F_2$ respectively. Point $P$ is a point in the coordinate plane, and $|\overrightarrow{OP}| = \frac{\sqrt{7}}{2}, \overrightarrow{PF_1} \cdot \overrightarrow{PF_2} = \frac{3}{4}$, where $O$ is the origin.
1. Find the equation of ellipse $C$.
2. A line $l$ passing through point $S(0, \frac{1}{3})$ with slope $k$ intersects the ellipse at points $A$ and $B$. Is there a fixed point $M$ on the y-axis such that the circle with diameter $AB$ always passes through this fixed point? If it exists, find the coordinates of point $M$; if not, explain why.

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