
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
$$
A_{m}=\left\{(x, y) \mid x, y \in \mathbf{Q}, x y \neq 0, \frac{x y}{m} \in \mathbf{Z}\right\}.
$$

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 $\beta(l)$ such that for any two points $M$ and $N$ on the line $l$, the following holds:
$$
f_{2016}(MN) \leq \lambda f_{2015}(MN) + \beta(l).

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