
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
Given three positive integers \(a_1, a_2, a_3\) with the following properties:
1. The numbers are distinct.
2. None of the numbers are perfect squares.
3. A triangle can be constructed using their square roots as side lengths, and this triangle has an integer area.

Find a procedure to generate an arbitrary number of fundamentally distinct such triples. Two triples are considered fundamentally distinct if the numbers of one cannot be obtained by multiplying the numbers of the other by the same integer.

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