
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
Consider a set $A_1$ of fractions where the numerators are integers in the interval $(0, m)$ (where $m > 1$ and $m \in \mathbb{N}$) and the denominator is $m$. The sum of all elements in this set is denoted as $a_1$. Consider another set $A_2$ of fractions not in $A_1$ where the numerators are integers in the interval $(0, m^2)$ and the denominator is $m^2$. The sum of all elements in this set is $a_2$. This process continues up to set $A_n$ where the numerators are integers in the interval $(0, m^n)$ and the denominator is $m^n$, and the sum of all elements is $a_n$. Find the sum $a_1 + a_2 + a_3 + \dots + a_n$.

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