
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
For positive integers \( n \), define \( d(n) = n - m^2 \), where \( m \) is the greatest integer such that \( m^2 \leq n \). Given a positive integer \( b_0 \), define a sequence \( b_i \) by taking \( b_{k+1} = b_k + d(b_k) \). For what values of \( b_0 \) do we have \( b_i \) constant for sufficiently large \( i \)?

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