
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
Non-intersecting circles \( S_1, S_2, \) and \( S_3 \) are consecutively inscribed in an angle of \( 60^\circ \). Find the area of the quadrilateral with vertices at the points of intersection with the sides of this angle of the common internal tangents of circles \( S_1 \) and \( S_2 \) and circles \( S_2 \) and \( S_3 \), given that the radius of circle \( S_2 \) is \( r \), and the difference in radii of circles \( S_3 \) and \( S_1 \) is \( a \).

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