
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
From the set of natural numbers \(\mathbf{N}\), select a sequence \(\left\{a_{n}\right\}\): Let \(a_{1}=1\), remove \(b_{1}=a_{1}+k\), where \(k\) is some natural number; next take \(a_{2}\) to be the smallest number in \(\mathbf{N}\) excluding \(a_{1}\) and \(b_{1}\), remove \(b_{2}=a_{2}+2k\); take \(a_{3}\) to be the smallest number in \(\mathbf{N}\) excluding \(a_{1}, a_{2}, b_{1}, b_{2}\), remove \(b_{3}=a_{3}+3k\), and continue in this manner to obtain the sequence \(\left\{a_{n}\right\}\). Find the expression for \(a_{n}\).

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