
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 "Detailed Explanation of the Nine Chapters Algorithm" by the mathematician Yang Hui of the Southern Song Dynasty, the concept of second-order arithmetic progression is discussed. It is different from the general arithmetic progression in that the differences between adjacent terms are not equal, but the differences between consecutive terms form an arithmetic progression. For example, the sequence $1$, $3$, $6$, $10$ has differences between adjacent terms that form a new sequence $2$, $3$, $4$, which is an arithmetic progression. Such sequences are called second-order arithmetic progressions. Given a second-order arithmetic progression $\{a_{n}\}$ where $a_{1}=2$, $a_{2}=3$, $a_{3}=5$, find $a_{100}$.

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