
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
If  $ n$  runs through all the positive integers, then  $ f(n) \equal{} \left \lfloor n \plus{} \sqrt {3n} \plus{} \frac {1}{2} \right \rfloor$  runs through all positive integers skipping the terms of the sequence  $ a_n \equal{} \left \lfloor \frac {n^2 \plus{} 2n}{3} \right \rfloor$ .

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