Magic Trees

Dohyun has one incubator, and it turns a tree into a magic tree.

There are $N$ trees in Dohyun's flower bed, numbered 1 to $N$. Some trees like other trees. Liking is not symmetric, and a tree can like itself. If tree $i$ likes tree $j$ then $L_{i,j} = 1$, and otherwise $L_{i,j} = 0$.

Every tree has two attributes. isMagical is True when the tree is a magic tree and False otherwise. isProtected is True when some other tree protects it and False otherwise. At the start, isMagical and isProtected are False for every tree.

Dohyun can pick one tree and change its isMagical from False to True. Turning one tree into a magic tree sets off the following changes in a chain.

• A tree $i$ whose isMagical is True protects every tree $j$ it likes. That is, if $L_{i,j} = 1$, the isProtected of tree $j$ becomes True.
• A tree $i$ whose isProtected is True protects every tree $j$ it likes. That is, if $L_{i,j} = 1$, the isProtected of tree $j$ becomes True.

The changes keep going until no tree changes any more. Once they are finished, Dohyun turns one more tree into a magic tree. He repeats this as many times as he wants.

Dohyun wants the largest possible number of trees that are magic and unprotected, meaning trees whose isMagical is True and whose isProtected is False. Write a program that finds that maximum.

The first line contains the number of trees $N$ ($1 \le N \le 50$).

Each of the next $N$ lines contains a string of length $N$ made of 0 and 1. The $j$-th character of the $i$-th line is $L_{i,j}$.

Print the largest possible number of trees that are magic and unprotected.

With two trees where only tree 1 likes tree 2, turn tree 1 into a magic tree. Tree 1 is then magic and unprotected.

With three trees where tree 1 likes tree 2 and tree 2 likes tree 3, turn tree 3 into a magic tree first, then turn tree 1 into a magic tree. Tree 1 protects tree 2 and tree 2 protects tree 3. Tree 1 is magic and unprotected, tree 2 is an ordinary tree and protected, and tree 3 is magic and protected.