
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
The side lengths \(a\), \(b\), and \(c\) of \(\triangle ABC\) (where \(a \leq b \leq c\)) simultaneously satisfy the following three conditions:
1. \(a\), \(b\), and \(c\) are all integers;
2. \(a\), \(b\), and \(c\) form a geometric sequence;
3. At least one of \(a\) or \(c\) is equal to 100.

Find all possible triplets \((a, b, c)\).

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