
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
Let  $T_1$  be an isosceles triangle with sides of length 8, 11, and 11.  Let  $T_2$  be an isosceles triangle with sides of length  $b$ , 1, and 1.  Suppose that the radius of the incircle of  $T_1$  divided by the radius of the circumcircle of  $T_1$  is equal to the radius of the incircle of  $T_2$  divided by the radius of the circumcircle of  $T_2$ .  Determine the largest possible value of  $b$ .  

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