
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
Define a function  $f$  as follows. For any positive integer  $i$ , let  $f(i)$  be the smallest positive integer  $j$  such that there exist pairwise distinct positive integers  $a,b,c,$  and  $d$  such that  $\gcd(a,b)$ ,  $\gcd(a,c)$ ,  $\gcd(a,d)$ ,  $\gcd(b,c)$ ,  $\gcd(b,d)$ , and  $\gcd(c,d)$  are pairwise distinct and equal to  $i, i+1, i+2, i+3, i+4,$  and  $j$  in some order, if any such  $j$  exists; let  $f(i)=0$  if no such  $j$  exists. Compute  $f(1)+f(2)+\dots +f(2019)$ .

*Proposed by Edward Wan*

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