
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
A sequence of integers \( a_{1}, a_{2}, a_{3}, \ldots \) is defined as follows: \( a_1 = 1 \), and for \( n \geq 1 \), \( a_{n+1} \) is the smallest integer greater than \( a_n \) such that \( a_i + a_j \neq 3a_k \) for any \( i, j, k \) in \( \{1, 2, \ldots, n+1\} \), not necessarily distinct. Determine \( a_{1008} \).

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