
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 \( n \) be a positive integer. Find the number of triples \(\langle a, b, c \rangle\) of positive integers that satisfy the following conditions:
(i) \(ab = n\);
(ii) \(1 \leqslant c \leqslant b\);
(iii) The greatest common divisor of \(a\), \(b\), and \(c\) is 1.

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