
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
From point \( P \) inside triangle \( \triangle ABC \), perpendiculars are drawn to sides \( BC \), \( CA \), and \( AB \) with the feet of the perpendiculars being \( D \), \( E \), and \( F \) respectively. Using \( AF \), \( BF \), \( BD \), \( CD \), \( CE \), \( AE \) as diameters, six semicircles are drawn outward. The areas of these six semicircles are denoted as \( S_{1}, S_{2}, S_{3}, S_{4}, S_{5}, S_{6} \). Given that \( S_{5} - S_{6} = 2 \) and \( S_{1} - S_{2} = 1 \), find \( S_{4} - S_{3} = \).

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