
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
We obtain the point \( P \) on the segment \( AB \) in the following steps:
1. Construct a segment \( BC = \frac{AB}{2} \) perpendicular to \( AB \) at \( B \).
2. Draw a line through \( C \) perpendicular to \( AC \), intersecting the line \( AB \) at point \( D \).
3. From \( D \), construct a segment \( DE \) equal in length and direction to \( AB \).
4. Draw a line through \( E \) perpendicular to \( AB \), intersecting the line \( AC \) at point \( F \).
5. Draw a parallel line to \( DF \) through \( C \), intersecting \( AB \) at point \( P \).

In what ratio does point \( P \) divide the segment \( AB \)?

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