
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
Triangle \(ABC\) has \(AB = 4\), \(BC = 3\), and a right angle at \(B\). Circles \(\omega_1\) and \(\omega_2\) of equal radii are drawn such that \(\omega_1\) is tangent to \(AB\) and \(AC\), \(\omega_2\) is tangent to \(BC\) and \(AC\), and \(\omega_1\) is tangent to \(\omega_2\). Find the radius of \(\omega_1\).

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