
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  $\mathbb{Q}$  be the set of rational numbers,  $\mathbb{Z}$  be the set of integers. On the coordinate plane, given positive integer  $m$ , define   $$ A_m = \left\{ (x,y)\mid x,y\in\mathbb{Q}, xy\neq 0, \frac{xy}{m}\in \mathbb{Z}\right\}. $$  For segment  $MN$ , define  $f_m(MN)$  as the number of points on segment  $MN$  belonging to set  $A_m$ .

Find the smallest real number  $\lambda$ , such that for any line  $l$  on the coordinate plane, there exists a constant  $\beta (l)$  related to  $l$ , satisfying: for any two points  $M,N$  on  $l$ ,   $$ f_{2016}(MN)\le \lambda f_{2015}(MN)+\beta (l) $$  

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