
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
Define  $a_k = (k^2 + 1)k!$  and  $b_k = a_1 + a_2 + a_3 + \cdots + a_k$ . Let \[\frac{a_{100}}{b_{100}} = \frac{m}{n}\] where  $m$  and  $n$  are relatively prime natural numbers. Find  $n - m$ .

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