I need to switch the position of two nodes in Graphviz

Question

Here is the code for my graphviz digraph I am trying to create. An example image of what this creates is below, and I want to switch node 'a' with 'e' and I need to switch node 'b' with 'd'. I have tried changing their position in code and other stuff too. Examples of what I currently have and what I desire are in links below.

digraph {

    graph[nodesep=1.75, ranksep=1.25, pad=".5", rankdir=LR];

    node[shape=circle];
    F[label=f];
    A[label=a];
    E[label=e];
    B[label=b];
    D[label=d];
    C[label=c];

    {rank=min; F};
    {rank=same; A; E};
    {rank=same; B; D};
    {rank=max; C};

    F -> {D};
    A -> {B D F};
    E -> {D F};
    B -> {C F};
    D -> {B};
    C -> {D};
}

Current Plot

Desired Plot

Answer 1

I corrected my code above and solved my own problem in combination with @JLH answer.

digraph {

    graph[nodesep=1, ranksep=1, pad=".5"];
    rankdir="LR";

    node[shape=circle];
    F[label=f];
    A[label=a];
    E[label=e];
    B[label=b];
    D[label=d];
    C[label=c];

    {rank=min; F};
    {rank=same; A E}
    {rank=same; B D}
    {rank=max; C};

    F -> {D};
    A -> {B D F};
    E -> {D F};
    B -> {C F}
    D -> {B};
    C -> {D};

    edge[style=invis];
    A -> E
    E -> B
    B -> D
}

The key to the answer is adding an invisible connection between E and B which pulls E back into the loop of things.

Answer 2

I got very close with this implementation, though still having a hard time aligning E horizontally with D. But this scheme gets the nodes in the order you specify using invisible directional links that enforce order according to rank.

digraph { 
rankdir=LR;
node[shape=circle];
nodesep=1.75; ranksep=1.25; pad=0.5;

{rank=same; rank=min; F; }
{rank = same; A -> E [style=invis]; }
{rank= same; B -> D [style=invis]; }
{rank=same; rank=max; C;}

F[label=f];
A[label=a];
E[label=e];
B[label=b];
D[label=d];
C[label=c];

A -> B;
A -> D;
A-> F;
B-> C;
B-> F;
C-> D;
D-> B;
F-> D;

nodesep=0.75;
E -> D;
E-> F;
}