We have three containers whose sizes are A pints, B pints and C pints, respectively, where A$,$ B$,$ C are all positive integers.

In the beginning, the A$-$pint container has a pints of water, the B$-$pint container has b pints of water, and the C$-$pint container

has c pints of water, where a$,$ b and c are non$-$negative integers. $($For example, we might have A $=10,$ B $=7,$ C $=4$ and

a $=0,$ b $=6,$ c $=4.)$ We are allowed one type of operation: pouring the contents of one container into another, stopping only

when the source container is empty or the destination container is full. We want to know if there is a sequence of pourings

that, in the end, leaves exactly k pints of water in any of the three containers. $($So the answer we seek is either YES or NO$.)$

$1)$ Model this problem as a graph problem: give a precise definition of the graph involved, and state the specific question

about this graph that needs to be answered.

$2)$ Design an efficient algorithm to solve the above problem following this format: $(1)$ explain the main idea of the algorithm, $(2)$ present its pseudo$-$code, $(3)$

prove its correctness, $(4)$ analyze its time complexity. It has to be most efficient time complexity.Problem $1$

We have three containers whose sizes are A pints, $B$ pints and $C$ pints, respectively, where $A,B,C$ are all positive integers.

In the beginning, the $A-$pint container has a pints of water, the $B-$pint container has $b$ pints of water, and the $C-$pint container

has $c$ pints of water, where $a,b$ and $c$ are non$-$negative integers. $($For example, we might have $A=10,B=7,C=4$ and

$a=0,b=6,c=4.)$ We are allowed one type of operation: pouring the contents of one container into another, stopping only

when the source container is empty or the destination container is full. We want to know if there is a sequence of pourings

that, in the end, leaves exactly $k$ pints of water in any of the three containers. $($So the answer we seek is either YES or NO$.)$

Your task:

Model this problem as a graph problem: give a precise definition of the graph involved, and state the specific question

about this graph that needs to be answered.

Design an efficient algorithm to solve the above problem.