
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
Find all real numbers \( c > 0 \) such that there exists a labeling of the lattice points in \(\mathbb{Z}^{2}\) with positive integers for which:
- only finitely many distinct labels occur, and
- for each label \(i\), the distance between any two points labeled \(i\) is at least \( c^{i} \).

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