
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
A subset $T$ of positive rational numbers is called complete if, from the fact that $\frac{p}{q} \in T$, it follows that both $\frac{p}{p+q}$ and $\frac{q}{p+q}$ are also in $T$.

Which positive rational numbers $r$ have the following property: if $r$ is contained in a complete set, then this set contains every rational number between 0 and 1?

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