
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 $a_1 = 1010$ and the second term is $a_2 = 1013.$ Furthermore, the values of the remaining terms are chosen so that
\[a_n + 2a_{n + 1} + a_{n + 2} = 3n + 2\] for all $n \ge 1.$ Determine $a_{1001}.$

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