
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 \( a \) and \( c \) be positive integers, and let \( b \) be a digit. Determine all triples \( (a, b, c) \) that satisfy the following conditions:

(1) \( (a, b b b \ldots)^{2}=c.777 \ldots \) (infinite decimals);

(2) \( \frac{c+a}{c-a} \) is an integer!

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