
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
130 trees (birches and lindens) are planted in a circle. Both types are present. Each tree has a sign stating: "Two different trees are growing nearby." It is known that all lindens have erroneous signs, and exactly one birch has an erroneous sign. How many birches could there be? Indicate all possible options.

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