
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 \( a_1 = 101 \) and the second term is \( a_2 = 102. \) Furthermore, the values of the remaining terms are chosen so that
\[ a_n + a_{n+1} + a_{n+2} = 2n + 1 \] for all \( n \ge 1. \) Determine \( a_{500}. \)

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