
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  $m$  and  $n$  be positive integers. Let  $S$  be the set of all points  $(x, y)$  with integer
coordinates such that  $1 \leq x,y \leq m+n-1$  and  $m+1 \leq x +y \leq 2m+n-1.$  Let  $L$  be the
set of the  $3m+3n-3$  lines parallel to one of  $x = 0, y = 0,$  or  $x + y = 0$  and passing through
at least one point in  $S$ . For which pairs  $(m, n)$  does there exist a subset  $T$  of  $S$  such that
every line in  $L$  intersects an odd number of elements of  $T$ ?

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