
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  $a_0=29$ ,  $b_0=1$  and   $$ a_{n+1} = a_n+a_{n-1}\cdot b_n^{2019}, \qquad b_{n+1}=b_nb_{n-1} $$   for  $n\geq 1$ . Determine the smallest positive integer  $k$  for which  $29$  divides  $\gcd(a_k, b_k-1)$  whenever  $a_1,b_1$  are positive integers and  $29$  does not divide  $b_1$ .

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