
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
One can define the greatest common divisor of two positive rational numbers as follows: for  $a$ ,  $b$ ,  $c$ , and  $d$  positive integers with  $\gcd(a,b)=\gcd(c,d)=1$ , write \[\gcd\left(\dfrac ab,\dfrac cd\right) = \dfrac{\gcd(ad,bc)}{bd}.\] For all positive integers  $K$ , let  $f(K)$  denote the number of ordered pairs of positive rational numbers  $(m,n)$  with  $m<1$  and  $n<1$  such that \[\gcd(m,n)=\dfrac{1}{K}.\] What is  $f(2017)-f(2016)$ ?

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