<s>[INST] Suppose that $x, y, z$ are three positive numbers that satisfy the equation $xyz=1, x+\frac{1}{z}=5$ and $y+\frac{1}{x}=29$. 
Suppose that $x, y, z$ are three positive numbers that satisfy the equation $xyz=1, x+\frac{1}{z}=5$ and $y+\frac{1}{x}=29$. Then $z+\frac{1}{y}=\frac{m}{n}$, where $m$ and $n$ are coprime. Find $m+n+1$.

I tried this, $$y=29-\frac{1}{x}$$ $$y=\frac{29x-1}{x}$$
And, $$x+\frac{1}{z}=5$$ $$z=\frac{1}{5-x}$$
What can I do next?
 [/INST] Just substitute the values of $y$ and $z$ in $xyz=1$.
$$\frac{x(29x-1)}{x(5-x)}=1$$
$$29x-1=5-x$$
$$x=\frac{1}{5}$$
$$y=\frac{29/5-1}{1/5}=24$$
$$z=\frac{1}{5-1/5}=\frac{5}{24}$$
$$z+\frac{1}{y}=\frac{5}{24}+\frac{1}{24}=\frac{1}{4}$$
Thus, $$m+n+1=1+4+1=6$$
</s>