
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 $a_1, a_2, \ldots$ be a sequence where $a_n= \frac{a_{n-1} + 2}{2}$ if $a_{n-1}$ is even and $a_n=3(a_{n-1}+2)+1$ if $a_{n-1}$ is odd. Determine how many positive integers $a_1 \le 3012$ ensure that $a_1$ is less than each of $a_2$, $a_3$, and $a_4$.

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