
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
For all positive integers  $n$ , let
\[f(n) = \sum_{k=1}^n\varphi(k)\left\lfloor\frac nk\right\rfloor^2.\] Compute  $f(2019) - f(2018)$ .  Here  $\varphi(n)$  denotes the number of positive integers less than or equal to  $n$  which are relatively prime to  $n$ .

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