
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
In the Cartesian plane is given a set of points with integer coordinate \[ T=\{ (x;y)\mid x,y\in\mathbb{Z} ; \ |x|,|y|\leq 20 ; \ (x;y)\ne (0;0)\} \] We colour some points of  $ T $  such that for each point  $ (x;y)\in T $  then either  $ (x;y) $  or  $ (-x;-y) $  is coloured. Denote  $ N $  to be the number of couples  $ {(x_1;y_1),(x_2;y_2)} $  such that both  $ (x_1;y_1) $  and  $ (x_2;y_2) $  are coloured and  $ x_1\equiv 2x_2 \pmod {41}, y_1\equiv 2y_2 \pmod {41} $ . Find the all possible values of  $ N $ .

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