
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
\( AB \) is a diameter of circle \( O \). \( X \) is a point on \( AB \) such that \( AX = 3BX \). Distinct circles \( \omega_1 \) and \( \omega_2 \) are tangent to \( O \) at \( T_1 \) and \( T_2 \) and to \( AB \) at \( X \). The lines \( T_1X \) and \( T_2X \) intersect \( O \) again at \( S_1 \) and \( S_2 \). What is the ratio \(\frac{T_1T_2}{S_1S_2}\)?

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