APMO 2011 Problem 4

Let $n$ be a fixed positive odd integer. Take $m+2$ distinct points $P_0,P_1,\ldots ,P_{m+1}$ (where $m$ is a non-negative integer) on the coordinate plane in such a way that the following three conditions are satisfied:

1. $P_0=(0,1),P_{m+1}=(n+1,n)$ , and for each integer $i,1\le i\le m$ , both $x$ - and $y$ - coordinates of $P_i$ are integers lying in between $1$ and $n$ ( $1$ and $n$ inclusive).
2. For each integer $i,0\le i\le m$ , $P_iP_{i+1}$ is parallel to the $x$ -axis if $i$ is even, and is parallel to the $y$ -axis if $i$ is odd.
3. For each pair $i,j$ with $0\le i<j\le m$ , line segments $P_iP_{i+1}$ and $P_jP_{j+1}$ share at most $1$ point. Determine the maximum possible value that $m$ can take.