
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
Two sequences  $\{a_i\}$  and  $\{b_i\}$  are defined as follows:  $\{ a_i \} = 0, 3, 8, \dots, n^2 - 1, \dots$  and  $\{ b_i \} = 2, 5, 10, \dots, n^2 + 1, \dots $ . If both sequences are defined with  $i$  ranging across the natural numbers, how many numbers belong to both sequences?

*Proposed by Isabella Grabski*

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