
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 *rank* of a rational number  $q$  is the unique  $k$  for which  $q=\frac{1}{a_1}+\cdots+\frac{1}{a_k}$ , where each  $a_i$  is the smallest positive integer  $q$  such that  $q\geq \frac{1}{a_1}+\cdots+\frac{1}{a_i}$ .  Let  $q$  be the largest rational number less than  $\frac{1}{4}$  with rank  $3$ , and suppose the expression for  $q$  is  $\frac{1}{a_1}+\frac{1}{a_2}+\frac{1}{a_3}$ .  Find the ordered triple  $(a_1,a_2,a_3)$ .

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