Five. (25 points) Given positive integers $a$, $b$, $c$ satisfying $a<b<c$, and $\frac{1}{a-1}+\frac{1}{b-1}+\frac{1}{c-1}=1$. Find the values of $a$, $b$, and $c$.

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Five. (25 points) Given positive integers $a$, $b$, $c$ satisfying $a<b<c$, and $\frac{1}{a-1}+\frac{1}{b-1}+\frac{1}{c-1}=1$. Find the values of $a$, $b$, and $c$.
Five, because $a\frac{1}{b-1}>\frac{1}{c-1} \text {. }$

Thus, $\frac{3}{a-1}>1 \Rightarrow a=2 \text {. }$

Therefore, $a=3$.
Similarly, $\frac{2}{b-1}>\frac{1}{2} \Rightarrow b<5$.
Since $a<b$, then $b=4$.
Substituting into $\frac{1}{a-1}+\frac{1}{b-1}+\frac{1}{c-1}=1$, we get $c=7$.
In summary, $a=3, b=4, c=7$.