CN-TST 2021 Problem 19

Let $n(\ge2)$ be a positive integer. Find the minimum $m$ , so that there exists $x_{ij}(1\le i ,j\le n)$ satisfying: (1)For every $1\le i ,j\le n, x_{ij}=max\{x_{i1},x_{i2},...,x_{ij}\}$ or $x_{ij}=max\{x_{1j},x_{2j},...,x_{ij}\}.$ (2)For every $1\le i \le n$ , there are at most $m$ indices $k$ with $x_{ik}=max\{x_{i1},x_{i2},...,x_{ik}\}.$ (3)For every $1\le j \le n$ , there are at most $m$ indices $k$ with $x_{kj}=max\{x_{1j},x_{2j},...,x_{kj}\}.$