
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
Circles of radius \( x \) and \( y \) touch a circle of radius \( R \), with the distance between the points of contact equal to \( a \). Calculate the length of the common tangent to the first two circles:

a) external, if both contacts are either external or internal simultaneously;

b) internal, if one contact is internal and the other is external.

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