
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 \( Q \) be the set of rational numbers and \( Z \) be the set of integers. In the coordinate plane, for a positive integer \( m \), define the set of points
\[ A_{m} = \left\{ (x, y) \mid x, y \in \mathbf{Q}, xy \neq 0, \frac{xy}{m} \in \mathbf{Z} \right\}. \]

For a line segment \( MN \), define \( f_{m}(MN) \) as the number of points on the line segment \( MN \) that belong to the set \( A_{m} \). Find the smallest real number \( \lambda \) such that for any line \( l \) in the coordinate plane, there exists a real number \( \beta(l) \) related to \( l \) such that for any two points \( M \) and \( N \) on the line \( l \), the inequality
\[ f_{2016}(MN) \leqslant \lambda f_{2015}(MN) + \beta(l) \]
holds.

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