
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
Let $f(x)$ and $g(x)$ be both increasing functions defined on the set of positive integers, and for any $k \in \mathbf{N}^{+}$, it holds that $f(k+1) + g(0) > f(k) + g(k-1)$. The sequence $\left\{a_{n}\right\}$ is formed by arranging all elements of the set $M = \{f(t) + g(s) \mid 0 \leqslant s < t, s, t \in \mathbf{N}\}$ in increasing order. Find the general term formula for the sequence $\left\{a_{n}\right\}$.

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