
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
A line $l$ passes through point $P(-1,0)$ and intersects the parabola $C: y^2=2px (p > 0)$ at points $A(x_1,y_1)$ and $B(x_2,y_2)$. If $\overrightarrow{OA} \cdot \overrightarrow{OB}=5$,

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

$(2)$ Determine whether there exists such a line $l$ such that both points $A$ and $B$ lie on a circle centered at $M(5,0)$. If it exists, find the equation of this line and the circle; if not, explain why.

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