
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 \(\omega_{1}\) and \(\omega_{2}\) with radii 10 and 13, respectively, are externally tangent at point \(P\). Another circle \(\omega_{3}\) with radius \(2 \sqrt{2}\) passes through \(P\) and is orthogonal to both \(\omega_{1}\) and \(\omega_{2}\). A fourth circle \(\omega_{4}\), orthogonal to \(\omega_{3}\), is externally tangent to \(\omega_{1}\) and \(\omega_{2}\). Compute the radius of \(\omega_{4}\).

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