
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
Compute the number of ordered triples of integers  $(a,b,c)$  between  $1$  and  $12$ , inclusive, such that, if   $$ q=a+\frac{1}{b}-\frac{1}{b+\frac{1}{c}}, $$   then  $q$  is a positive rational number and, when  $q$  is written in lowest terms, the numerator is divisible by  $13$ .

*Proposed by Ankit Bisain*

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