Find the first n pairs in a list whose difference is the nearest to some number

This is my list:

list = {2.6135182051634342, -2.6690957180727297, 
   0.26060239174734157, -0.4182243295792278};

I wanna gather it every two number,which make this group's diffrence's absolute value is nearest 3.14.

We can see this distance martix.

 DistanceMatrix[list, 
      DistanceFunction -> (First[Abs[#1 - #2]] &)] // MatrixForm

As we see.The $(1st,4th)$ and $(2nd,3rd)$ is we want to group.Implies we result is {{2.6135182051634342,-0.4182243295792278},{-2.6690957180727297,0.26060239174734157}}

But this code doesn't work,becuase the Nearest's first parameter cannot be a matrix.

mat = DistanceMatrix[list, 
   DistanceFunction -> (First[Abs[#1 - #2]] &)];
Position[mat, Nearest[mat, 3.14, 2]]

Or there are other more better method to do this?

---

Let's extended this topic.Assumption we have a list2 like this.But wanna group it every two element that make its sum nearest 100.So how to do it?

SeedRandom[2016317];
list2 = RandomSample[Range@100, 20];

DistanceMatrix[list, 
  DistanceFunction -> (First[Plus[#1, #2]] &)] // MatrixForm

Answer

f[l_List, nbr_Integer, near_Real] := Module[{k, f},
  k = Subsets[Range@Length@l, {2}]; 
  f = Nearest[# -> Range@Length@#] &[EuclideanDistance @@ l[[#]] & /@ k];
  k[[f[near, nbr]]]
  ]


list = {2.6135182051634342, -2.6690957180727297, 
       0.26060239174734157, -0.4182243295792278};

f[list, 2, N@Pi]

(* {{1, 4}, {2, 3}} *)

Your other example:

SeedRandom[2016317];

list2 = RandomSample[Range@100, 20];
f[list2, 2, 100.]

(* {{14, 18}, {18, 20}} *)