
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
Consider an isosceles triangle \( T \) with base 10 and height 12. Define a sequence \( \omega_1, \omega_2, \ldots \) of circles such that \( \omega_1 \) is the incircle of \( T \) and \( \omega_{i+1} \) is tangent to \( \omega_i \) and both legs of the isosceles triangle for \( i > 1 \).
Find the radius of \( \omega_1 \).

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