
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
Choose positive integers \( b_{1}, b_{2}, \ldots \) satisfying
\[
1=\frac{b_{1}}{1^{2}}>\frac{b_{2}}{2^{2}}>\frac{b_{3}}{3^{2}}>\frac{b_{4}}{4^{2}}>\cdots
\]
and let \( r \) denote the largest real number satisfying \( \frac{b_{n}}{n^{2}} \geq r \) for all positive integers \( n \). What are the possible values of \( r \) across all possible choices of the sequence \( \left( b_{n} \right) \)?

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