
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
Let the focus of the parabola \( C: y^2 = 4x \) be \( F \). A line \( L \) passing through \( F \) intersects \( C \) at points \( P \) and \( Q \). Let \( L \) intersect the directrix of the parabola \( C \) at point \( M \), and \( \overrightarrow{FM} = 3 \overrightarrow{FP} \). Find \( |\overrightarrow{FP}| \).

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