Bessie the cow is just learning how to convert numbers between different bases, but she keeps making errors since she cannot easily hold a pen between her two front hooves. Whenever Bessie converts a number to a new base and writes down the result, she always writes one of the digits wrong. For example, if she converts the number $14$ into binary (i.e., base $2$), the correct result should be `1110`, but she might instead write down `0110` or `1111`. Bessie never accidentally adds or deletes digits, so she might write down a number with a leading digit of `0` if this is the digit she gets wrong. Given Bessie's output when converting a number $N$ into base $2$ and base $3$, please determine the correct original value of $N$ (in base $10$). You can assume $N$ is at most $1$ billion, and that there is a unique solution for $N$. Please feel welcome to consult any online reference you wish regarding base-$2$ and base-$3$ numbers, if these concepts are new to you.

Line $1$: The base-$2$ representation of $N$, with one digit written incorrectly. Line $2$: The base-$3$ representation of $N$, with one digit written incorrectly.

Line $1$: The correct value of $N$.

Based on the given information, when Bessie incorrectly converts $N$ into base $2$, she writes down `1010`, and when she incorrectly converts $N$ into base $3$, she writes down `212`. The correct value of $N$ is $14$, which is represented as `1110` in base $2$ and `112` in base $3$.