
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 set  $S$  of rational numbers is ordered in a tree-diagram in such a way that each rational number  $\frac{a}{b}$  (where  $a$  and  $b$  are coprime integers) has exactly two successors:  $\frac{a}{a+b}$  and  $\frac{b}{a+b}$ . How should the initial element  be selected such that this tree contains the set of all rationals  $r$  with  $0 < r < 1$ ? Give a procedure for determining the level of a rational number  $\frac{p}{q}$  in this tree.

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