
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
Observe the following equations:
$$\frac{1}{3} + \frac{2}{3} = 1$$
$$\frac{7}{3} + \frac{8}{3} + \frac{10}{3} + \frac{11}{3} = 12$$
$$\frac{16}{3} + \frac{17}{3} + \frac{19}{3} + \frac{20}{3} + \frac{22}{3} + \frac{23}{3} = 39$$
$$\dots$$

Then, when $m < n$ and $m,n \in \mathbb{N}$, find the expression for 
$$\frac{3m+1}{3} + \frac{3m+2}{3} + \frac{3m+4}{3} + \frac{3m+5}{3} + \dots + \frac{3n-2}{3} + \frac{3n-1}{3}$$
in terms of $m$ and $n$.

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