
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
Given: The sequence $\{a_n\}$ is an arithmetic sequence with the first term being $1$, and the common difference is not zero. The geometric sequence $\{b_n\}$ has its first three terms as $a_1$, $a_2$, and $a_6$ respectively.

(1) Find the general formula for the sequence $\{a_n\}$; (2) If $b_1 + b_2 + \cdots + b_k = 85$, find the value of the positive integer $k$.

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