
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  $ABC$  be an equilateral triangle with side 3. A circle  $C_1$  is tangent to  $AB$  and  $AC$ .
A circle  $C_2$ , with a radius smaller than the radius of  $C_1$ , is tangent to  $AB$  and  $AC$  as well as externally tangent to  $C_1$ .
Successively, for  $n$  positive integer, the circle  $C_{n+1}$ , with a radius smaller than the radius of  $C_n$ , is tangent to  $AB$  and  $AC$  and is externally tangent to  $C_n$ .
Determine the possible values for the radius of  $C_1$  such that 4 circles from this sequence, but not 5, are contained on the interior of the triangle  $ABC$ .

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