
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
In a certain sequence the first term is \( a_1 = 502 \) and the second term is \( a_2 = 503 \). Furthermore, the terms are chosen so that \( a_n + 2a_{n+1} + 3a_{n+2} = 3n \) for all \( n \geq 1 \). Determine \( a_{1000} \).

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