
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
Let \(a\) and \(b\) be positive integers. How many non-negative integers \(n\) are there such that

$$
\begin{aligned}
& \left\lfloor \frac{n}{ab} \right\rfloor + \left\lfloor \frac{n+b}{ab} \right\rfloor + \left\lfloor \frac{n+2b}{ab} \right\rfloor + \ldots + \left\lfloor \frac{n+(a-1)b}{ab} \right\rfloor = \\
& = \left\lfloor \frac{n}{ab} \right\rfloor + \left\lfloor \frac{n+a}{ab} \right\rfloor + \left\lfloor \frac{n+2a}{ab} \right\rfloor + \ldots + \left\lfloor \frac{n+(b-1)a}{ab} \right\rfloor ?
\end{aligned}
$$

(\(\lfloor x \rfloor\) denotes the floor function, which is the greatest integer less than or equal to \(x\)).

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