
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 point $P(1, m)$ is on the parabola $C$: $y^2 = 2Px (P > 0)$, and $F$ is the focus with $|PF| = 3$.
(I) Find the equation of the parabola $C$;
(II) Line $l$ passes through point $T(4, 0)$ and intersects parabola $C$ at points $A$ and $B$, with $O$ being the origin.
(i) Find the value of $\overrightarrow{OA} \cdot \overrightarrow{OB}$;
(ii) A circle with center $A$ and radius $|AT|$ intersects the $y$-axis at points $M$ and $N$. Find the area of $\triangle MNF$.

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