
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
Let  $a_n$  and  $b_n$  to be two sequences defined as below: $i)$   $a_1 = 1$  $ii)$   $a_n + b_n = 6n - 1$  $iii)$   $a_{n+1}$  is the least positive integer different of  $a_1, a_2, \ldots, a_n, b_1, b_2, \ldots, b_n$ .

Determine  $a_{2009}$ .

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