
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 the line $y=\frac{1}{2}x-1$ intersects the parabola $C:x^{2}=-2py\left(p \gt 0\right)$ at points $M(x_{M}$,$y_{M})$ and $N(x_{N}$,$y_{N})$, and $(x_{M}+1)(x_{N}+1)=-8$.<br/>$(1)$ Find the equation of $C$;<br/>$(2)$ If the line $y=kx-\frac{3}{2}({k≠0})$ intersects $C$ at points $A$ and $B$, where point $A$ is symmetric to point $A'$ with respect to the $y$-axis, is the line $A'B$ passing 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}
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