
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 $b_1, b_2, \ldots$ be a sequence defined by the rule $b_n = \frac{b_{n-1}}{3}$ if $b_{n-1}$ is divisible by 3, and $b_n = 2b_{n-1} + 1$ if $b_{n-1}$ is not. Determine how many positive integers $b_1 \leq 2023$ are such that $b_1$ is less than each of $b_2$, $b_3$, and $b_4$.

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