The value of xyz is 55 or `(343)/(55)` according as the series `a,x,y,z,b` is an AP or HP. Find the values of a and b given that they are positive integers.

Explanation

To solve the problem, we need to find the values of \( a \) and \( b \) given that the series \( a, x, y, z, b \) can either be in Arithmetic Progression (AP) or Harmonic Progression (HP), and the product \( xyz \) is either \( 55 \) or \( \frac{343}{55} \).

### Step 1: Case 1 - AP

Assuming \( a, x, y, z, b \) are in AP, we can express \( x, y, z \) in terms of \( a \) and \( b \).

1. The common difference \( d \) can be expressed as:
\[
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