
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
In $\triangle ABC$, the side lengths $a, b, c$ ($a \leq b \leq c$) simultaneously satisfy the following three conditions:
(i) $a, b, c$ are all integers;
(ii) $a, b, c$ form a geometric progression;
(iii) At least one of $a$ or $c$ is equal to 100.

Find all possible solutions for the triple $(a, b, c)$.

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