
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 that the vertex of the parabola $C$ is at the origin, the axis of symmetry is the coordinate axis, and its directrix passes through the left focus of the hyperbola $\frac{x^2}{2}-\frac{y^2}{2}=1$.  
$(1)$ Find the equation of the parabola $C$;  
$(2)$ If a line passing through the point $P(3,1)$ with a slope of $1$ intersects the parabola $C$ at points $M$ and $N$, find $|MN|$.

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