In a space with dimensions width: $,$ height: $,$ and depth: $,$ boxes are stacked with a width, height, and depth of $1$ each $($a total of Boxes$).$ Each box is marked either $\text{'}\text{'}$ or $\text{'}\text{'}.$ Now, considering all the pairs of coordinates, what is the maximum number of boxes stacked continuously along the $-$axis marked $\text{'}$A$\text{'}?$

For illustration, we represent $\text{'}\text{'}$ as yellow, and $\text{'}\text{'}$ as blue.

Example$)$ The maximum number of continuously stacked $\text{'}\text{'}$s along the Z$-$axis, which is yellow, corresponds to boxes at coordinate $.$

The order of inputs for the marks on the boxes is: $($red arrow in right figure$)($X$=0,$Y$=0,$Z$=0)->($X$=0,$Y$=1,$Z$=0)->($X$=0,$Y$=2,$Z$=0)->($X$=1,$Y$=0,$Z$=0)->($X$=1,$Y$=1,$Z$=0)->($X$=1,$Y$=2,$Z$=0)->...->($X$=2,$Y$=2,$Z$=0)->($X$=0,$Y$=0,$Z$=1)->...->($X$=2,$Y$=2,$Z$=2)$ The input consists of the dimensions of the space $($width$,$ height, depth$)$ and the marks on the boxes given as a string.

Input is provided by using cin $,$ and the result is displayed by using cout in Problem $1.$ In other words, the text is provided by keyboard typing.

The first line contains three integers X Y Z separated by space

After you receive the input X Y Z $,$ validate the input using the function is$\_$valid $.$

You have to utilize the function is$\_$valid in the main function. Otherwise, you will receive a deduction of $5$ points.

Implement the function is$\_$valid with the following specifications:

The function is$\_$valid receives three arguments X$,$Y$,$Z $.$

It verifies if these values, X Y Z $,$ are within the range of $ X$,$ Y$,$ Z $$.$

If inputs are valid, return $1.$ Otherwise, return $0.$ If inputs X Y Z are invalid, receive an input text X Y Z again until it$$s valid.

If inputs are valid, receive the next inputs $($array of characters$).$ If input X Y Z is valid, get an input text whose length is the range of is $.$

After receiving the input text, validate it$.$

$1.$ The length of the text equals $.$

$2.$ Whether the text contains only $$A$$ or $$B$.$ If it contains other characters such as $$C$$ or $$Q$,$ it is not valid. If the input is invalid, receive the text again, until it$$s valid.

Output Format

Display the maximum number of boxes stacked continuously along the Z$-$axis marked $\text{'}$A$\text{'}$ by using

cout$$