
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
Create a fraction set $A_1$ with numerators being the integers in the interval $(0, m)$ ($m > 1$, $m \in \mathbb{N}$) and the denominator being $m$, with the sum of all elements denoted by $a_1$. Construct another fraction set $A_2$ with numerators being the integers in the interval $(0, m^2)$ ($m > 1$, $m \in \mathbb{N}$) and the denominator being $m^2$, such that $A_2$ contains no elements from set $A_1$, with the sum of all elements denoted by $a_2$. Continuing in this manner, create the fraction set $A_n$ using the integers in the interval $(0, m^n)$ as numerators ($m > 1$, $m \in \mathbb{N}$) and $m^n$ as the denominator, ensuring that $A_n$ includes no elements from $A_1$, $A_2$, ..., $A_{n-1}$, with the sum of all elements denoted by $a_n$. Find the sum $a_1 + a_2 + ... + a_n$.

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