4. The workloads of projects A, B, and C are in the ratio of $1: 2: 3$, undertaken by teams 甲, 乙, and 丙 respectively, starting simultaneously. After several days, the work completed by 甲 is half of the work not completed by 乙, the work completed by 乙 is one-third of the work not completed by 丙, and the work completed by 丙 is equal to the work not completed by 甲. What is the ratio of the work efficiencies of teams 甲, 乙, and 丙?
4. Solution: Let the work efficiencies of the three teams be $\frac{1}{a}$, $\frac{1}{b}$, and $\frac{1}{c}$, and the workloads of the three projects be $1$, $2$, and $3$, respectively. Let the number of days be $k$ days.

Then after $k$ days, the work completed by Team A is $\frac{k}{a}$, and the remaining work is $1-\frac{k}{a}$. The work completed by Team B is $\frac{k}{b}$, and the remaining work is $2-\frac{k}{b}$. The work completed by Team C is $\frac{k}{c}$, and the remaining work is $3-\frac{k}{c}$. Therefore, we have:
$$
\left\{\begin{array}{l}
\frac{k}{a}=\frac{1}{2}\left(2-\frac{k}{b}\right) \\
\frac{k}{b}=\frac{1}{3}\left(3-\frac{k}{c}\right) \\
\frac{k}{c}=1-\frac{k}{a}
\end{array}\right.
$$

From this, we can derive: $\frac{1}{a}+\frac{1}{2 b}=\frac{1}{b}+\frac{1}{3 c}=\frac{1}{c}+\frac{1}{a}$
Thus, we get: $\frac{1}{c}=\frac{1}{2 b}$, i.e., $\frac{1}{b}: \frac{1}{c}=2$,
$\frac{1}{a}+\frac{1}{2 b}=\frac{1}{b}+\frac{1}{6 b}$, which leads to $\frac{1}{a}=\frac{6+1-3}{6 b}=\frac{2}{3 b}$, i.e., $\frac{1}{a}: \frac{1}{b}=\frac{2}{3}$,
$$
\therefore \frac{1}{a}: \frac{1}{b}: \frac{1}{c}=4: 6: 3
$$