**A**,

**B**,

**C**,

**D**and

**E**, with

**A**<

**B**<

**C**<

**D**<

**E**, are such that:

(i)

**A**,

**B**and

**C**(in this order) are in

**harmonic sequence**, and:

(ii)

**B**,

**C**and

**D**(in this order) are in geometric sequence, and:

(iii)

**C**,

**D**and

**E**(in this order) are in arithmetic sequence.

Determine the minimum value of (

**E**-

**A**) such that there are precisely three quintuplets (

**A**,

**B**,

**C**,

**D**,

**E**) that satisfy all the given conditions.