
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 \(a_n = 5a_{n-1} + 1\) if \(a_{n-1}\) is odd. For how many positive integers \(a_1 \le 1000\) 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