
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
In the plane Cartesian coordinate system \( xOy \), the focus of the parabola \( \Gamma: y^2 = 2px \) (with \( p > 0 \)) is \( F \). A tangent is drawn to \( \Gamma \) at a point \( P \) different from \( O \), and this tangent intersects the y-axis at point \( Q \). Given that \( |FP| = 2 \) and \( |FQ| = 1 \), find the dot product of the vectors \( \overrightarrow{OP} \) and \( \overrightarrow{OQ} \).

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