
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 an infinite sequence of positive integers such that for integers \( n > 2 \), \( a_n = 3a_{n-1} - 2a_{n-2} \). How many such sequences \( \{a_n\} \) are there such that \( a_{2010} \leq 2^{2012} \)?

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