
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 $F\_1F\_2$, and point $P\left( {{x}\_{0}},{{y}\_{0}} \right)$ is a point in the coordinate plane such that $\left| OP \right|=5$, and $\overrightarrow{PF_{1}}\cdot \overrightarrow{PF_{2}}=16(O$ is the coordinate origin$).

(1) Find the equation of ellipse $C$.
(2) Does a fixed point $M$ exist on the $y$-axis such that a moving straight line $l$ passing through point $S\left( 0,-1 \right)$ with a slope of $k$ intersects the ellipse at points $A$ and $B$, and a circle with $AB$ as its diameter always passes through this point? If it exists, find the coordinates of point $M$; if not, explain the reason.

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