
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
The side length of the square \(ABCD\) is 28. On side \(CD\), point \(E\) is marked, and inside the square, point \(P\) is such that \(PE \perp CD\) and \(AP = PB = PE\). Find the length of the segment \(AP\). If necessary, round the answer to the nearest hundredth.

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