
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 determined by the rule \( a_n = \frac{a_{n-1}}{2} \) if \( a_{n-1} \) is even and \( a_n = 3a_{n-1} + 1 \) if \( a_{n-1} \) is odd. For how many positive integers \( a_1 \le 3000 \) is it true that \( a_1 \) is less than each of \( a_2 \), \( a_3 \), and \( a_4 \)?

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