
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 coordinate origin $O$, the axis of symmetry is the $x$-axis, the focus is $F$, and a point $A$ on the parabola has an x-coordinate of $2$, with $\overrightarrow{FA} \cdot \overrightarrow{OA} = 10$.

1. Find the equation of this parabola $C$.
2. A line $l$ passing through the point $(4, 0)$ intersects the parabola $C$ at points $M$ and $N$. Find the value of $\overrightarrow{OM} \cdot \overrightarrow{ON}$.

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