
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
The sequence an is defined by  $a_0 = 4, a_1 = 1$  and the recurrence formula  $a_{n+1} = a_n + 6a_{n-1}$ . The sequence  $b_n$  is given by
\[b_n=\sum_{k=0}^n \binom nk a_k.\]
Find the coefficients  $\alpha,\beta$   so that  $b_n$  satisfies the recurrence formula  $b_{n+1} = \alpha b_n + \beta b_{n-1}$ . Find the explicit form of  $b_n$ .

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