Sarthak can fill a sandpit with sand in $14\text{days}$ while Vivan takes $35\text{days}$ to fill it. Ali can empty all of the sand from a filled sandpit in $12\text{days}$. If all three start working when the pit is empty, in how many days will the sandpit be full again?

Let's determine the daily work efficiency of each individual.
* Sarthak's daily filling efficiency = $+\frac{1}{14}$
* Vivan's daily filling efficiency = $+\frac{1}{35}$
* Ali's daily emptying efficiency = $-\frac{1}{12}$

When all three work together, the net work done per day is:

$$\text{Net Work per day}=\frac{1}{14}+\frac{1}{35}-\frac{1}{12}$$

Find the LCM of $14,35$, and $12$:

* $14=2\times 7$
* $35=5\times 7$
* $12=2^2\times 3$

$$\text{LCM}=2^2\times 3\times 5\times 7=420$$

Now convert the fractions:

$$\text{Net Work per day}=\frac{30+12-35}{420}$$

$$\text{Net Work per day}=\frac{42-35}{420}=\frac{7}{420}=\frac{1}{60}$$

Since $\frac{1}{60}$ of the sandpit is filled in one day, the total time required to completely fill the sandpit is:

$$\text{Total Time}=60\text{days}$$