
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
Two circles are concentric. A chord \( AC \) of the outer circle touches the inner circle at \( Q \). \( P \) is the midpoint of \( AQ \). A line through \( A \) intersects the inner circle at \( R \) and \( S \). The perpendicular bisectors of \( PR \) and \( CS \) meet at \( T \) on the line \( AC \). What is the ratio \( AT/TC \)?

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