
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 a parabola $y^{2}=2px (p > 0)$ is intersected by the line $y=x-1$ with a chord length of $2\sqrt{6}$.

$(1)$ Find the equation of the parabola.

$(2)$ With this chord as the base and a point $P$ on the $x$-axis as the vertex, form a triangle. When the area of this triangle is $5\sqrt{3}$, find the coordinates of point $P$.

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