
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 have radius  $2$  and  $3$ , and the distance between their centers is  $10$ . Let  $E$  be the intersection of their two common external tangents, and  $I$  be the intersection of their two common internal tangents. Compute  $EI$ .

(A *common external tangent* is a tangent line to two circles such that the circles are on the same side of the line, while a *common internal tangent* is a tangent line to two circles such that the circles are on opposite sides of the line).

*Proposed by Connor Gordon)*

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