
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
A sequence \( b_1, b_2, b_3, \dots \) is defined recursively by \( b_1 = 2, b_2 = 2, \) and for \( k \ge 3, \)
\[
b_k = \frac{1}{2} b_{k - 1} + \frac{1}{3} b_{k - 2}.
\]
Evaluate \( b_1 + b_2 + b_3 + \dotsb. \)

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