
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 in triangle \( \triangle ABC \), lines \( AD \), \( BE \), and \( CF \) intersect at point \( P \), with \( \frac{AF}{BF} = \frac{AE}{CE} \) and \( AD \perp BC \), and lines through point \( C \) parallel to \( AB \) intersect the extension of \( BE \) at point \( G \), with \( PE = 1 \) and \( EG = 2 \), find the value of \( BP \).

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