You are a helpful assistant that can solve the given question step by step with the help of the wikipedia search tool and python interpreter tool. Given a question, you need to first think about the reasoning process in the mind and then provide the answer. During thinking, you can invoke the wikipedia search tool to search and python interpreter tool to calculate the math problem for fact information about specific topics if needed. The reasoning process and answer are enclosed within <think> </think> and <answer> </answer> tags respectively, and the search query and result are enclosed within <search> </search> and <result> </result> tags respectively. For example, <think> This is the reasoning process. </think> <search> search query here </search> <result> search result here </result> <think> This is the reasoning process. </think> <python> python code here </python> <result> python interpreter result here </result> <think> This is the reasoning process. </think> <answer> The final answer is \[ oxed{answer here} \] </answer>. In the last part of the answer, the final exact answer is enclosed within oxed{} with latex format.
system
Given an integer \( n > 1 \) and a \( n \times n \) grid \( ABCD \) containing \( n^2 \) unit squares, each unit square is colored by one of three colors: Black, white, and gray. A coloring is called *symmetry* if each unit square with its center on diagonal \( AC \) is colored gray, and every pair of unit squares that are symmetric about \( AC \) must be colored either both black or both white. In each gray square, the label is \( 0 \); in a white square, a positive integer is labeled; and in a black square, a negative integer is labeled. A label is called \( k \)-*balance* (with \( k \in \mathbb{Z}^+ \)) if it satisfies the following requirements:

1. Each pair of unit squares that are symmetric about \( AC \) are labeled with the same integer from the closed interval \([-k, k]\).
2. If a row and a column intersect at a square colored black, then the set of positive integers in that row and the set of positive integers in that column must be distinct. If they intersect at a square colored white, then the set of negative integers in that row and the set of negative integers in that column must be distinct.

For \( n = 5 \), determine the minimum value of \( k \) such that there exists a \( k \)-balance label for the specified grid.

Additionally, for \( n = 2017 \), find the least value of \( k \) such that there is always a \( k \)-balance label for a symmetry coloring.
user
2017
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Given an integer \( n > 1 \) and a \( n \times n \) grid \( ABCD \) containing \( n^2 \) unit squares, each unit square is colored by one of three colors: Black, white, and gray. A coloring is called *symmetry* if each unit square with its center on diagonal \( AC \) is colored gray, and every pair of unit squares that are symmetric about \( AC \) must be colored either both black or both white. In each gray square, the label is \( 0 \); in a white square, a positive integer is labeled; and in a black square, a negative integer is labeled. A label is called \( k \)-*balance* (with \( k \in \mathbb{Z}^+ \)) if it satisfies the following requirements:

1. Each pair of unit squares that are symmetric about \( AC \) are labeled with the same integer from the closed interval \([-k, k]\).
2. If a row and a column intersect at a square colored black, then the set of positive integers in that row and the set of positive integers in that column must be distinct. If they intersect at a square colored white, then the set of negative integers in that row and the set of negative integers in that column must be distinct.

For \( n = 5 \), determine the minimum value of \( k \) such that there exists a \( k \)-balance label for the specified grid.

Additionally, for \( n = 2017 \), find the least value of \( k \) such that there is always a \( k \)-balance label for a symmetry coloring.
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