
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 point P (1, m) on the parabola C: $y^2 = 2px$ ($p > 0$), where F is the focus and $PF = 3$.

(1) Find the equation of the parabola C.
(2) A line l passing through point T (4, 0) intersects the parabola C at points A and B. O is the origin. Find the value of $\overrightarrow{OA} \cdot \overrightarrow{OB}$.

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