
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 $\frac{\sqrt{2}}{2}$, its left and right foci are $F\_1$ and $F\_2$, respectively. Point $P(x\_0, y\_0)$ is a point in the coordinate plane such that $|OP|= \frac{\sqrt{7}}{2}, \overrightarrow{PF\_1} \cdot \overrightarrow{PF\_2} = \frac{3}{4} (O$ is the coordinate origin$)$.
(1) Find the equation of the ellipse $C$;
(2) A moving straight line $l$ passes through the point $S(0, -\frac{1}{3})$ with a slope of $k$. It intersects the ellipse at points $A$ and $B$. Does there exist a fixed point $M$ on the $y$-axis such that a circle with $AB$ as its diameter always passes through this point? If so, find the coordinates of $M$. If not, explain why.

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