
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
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.

Present the answer in LaTex format: \boxed{Your answer}
user