A alone can complete a work in $200$ days, while B alone can complete the same work in $100$ days. In each three-day cycle, on the first day both A and B work together, on the second day only A works, and on the third day only B works. This cycle continues until the work is completed. How many days will they take together to complete the work?

Step 1: Determine Total Work and Efficiencies
Let the total work be the LCM of $200$ and $100$, which is $200$ units.

* Efficiency of A = $\frac{200}{200}=1$ unit/day
* Efficiency of B = $\frac{200}{100}=2$ units/day

Step 2: Analyze the 3-day cyclic pattern

* Day 1: Both A and B work = $1+2=3$ units
* Day 2: Only A works = $1$ unit
* Day 3: Only B works = $2$ units

Total work done in 1 cycle ($3$ days) = $3+1+2=6$ units.

Step 3: Calculate the number of completed cycles
We need to complete a total of $200$ units of work.
Number of complete cycles = $\lfloor \frac{200}{6}\rfloor =33$ cycles.

Work completed in $33$ cycles = $33\times 6=198$ units.
Time taken for $33$ cycles = $33\times 3=99$ days.

Step 4: Deal with the remaining work
Remaining work = $200-198=2$ units.

On the $100^{\text{th}}$ day (which is Day 1 of the $34^{\text{th}}$ cycle), both A and B work together with a combined efficiency of $3$ units/day.
Time taken to finish the remaining $2$ units = $\frac{\text{Remaining Work}}{\text{Combined Efficiency}}=\frac{2}{3}$ day.

Total Time Taken:

$$\text{Total Time}=99+\frac{2}{3}=99\frac{2}{3}\text{days}$$