
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 said to be *orthogonal* if they intersect in two points, and their tangents at either point of intersection are perpendicular. 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  $2\sqrt2$  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}
user