Let $a=\frac{x}{y}+\frac{y}{z}+\frac{z}{x}$, let $b=\frac{y}{x}+\frac{z}{y}+\frac{x}{z}$ and let $c=\left(\frac{x}{y}+\frac{y}{z}\right)\left(\frac{y}{z}+\frac{z}{x}\right)\left(\frac{z}{x}+\frac{x}{y}\right)$. The value of $|ab-c|$ is:

Step 1: Let $u=\frac{x}{y}$, $v=\frac{y}{z}$, and $w=\frac{z}{x}$. Note that the product of these variables is:

Step 2: We can express $a$ and $b$ in terms of $u,v,w$:

since $uvw=1$.

Step 3: Now we express $c$ in terms of $u,v,w$:

Step 4: Using the standard algebraic identity for the product of sums of three variables:

Step 5: Substituting $a,b,$ and $uvw=1$ into the identity:

Step 6: This gives:

Thus, the value of $|ab-c|$ is $1$.