
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
Given $p$ and $q$ as two coprime integers, establish that $\sum_{k=0}^{q-1}\left\lfloor\frac{k p}{q}\right\rfloor=\frac{(p-1)(q-1)}{2}$.

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