
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
Point \( P \) is inside equilateral \( \triangle ABC \). Points \( Q \), \( R \), and \( S \) are the feet of the perpendiculars from \( P \) to \( \overline{AB} \), \( \overline{BC} \), and \( \overline{CA} \), respectively. Given that \( PQ = 3 \), \( PR = 4 \), and \( PS = 5 \), what is the side length \( AB \)?

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