
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
In triangle \( PQR \), the angle \( \angle QPR \) is \( 60^{\circ} \). Perpendiculars are drawn from vertices \( P \) and \( R \) to the sides \( QR \) and \( PQ \) respectively. The point of intersection of these perpendiculars is 1 unit away from vertices \( P \) and \( Q \). Find the sides of triangle \( PQR \).

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