
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 the diagram, the circles with centers \( B \) and \( D \) have radii 1 and \( r \), respectively, and are tangent at \( C \). The line through \( A \) and \( D \) passes through \( B \). The line through \( A \) and \( S \) is tangent to the circles with centers \( B \) and \( D \) at \( P \) and \( Q \), respectively. The line through \( A \) and \( T \) is also tangent to both circles. Line segment \( ST \) is perpendicular to \( AD \) at \( C \) and is tangent to both circles at \( C \).

(a) There is a value of \( r \) for which \( AS = ST = AT \). Determine this value of \( r \).

(b) There is a value of \( r \) for which \( DQ = QP \). Determine this value of \( r \).

(c) A third circle, with center \( O \), passes through \( A, S \) and \( T \), and intersects the circle with center \( D \) at points \( V \) and \( W \). There is a value of \( r \) for which \( OV \) is perpendicular to \( DV \). Determine this value of \( r \).

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