
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 plane region
$$
M=\left\{(x, y) \left\lvert\,\left\{\begin{array}{l}
0 \leqslant y \leqslant 2-x, \\
0 \leqslant x \leqslant 2
\end{array}\right\}\right.\right.
$$

If we select $k$ points randomly within $M$, and we can always divide these $k$ points into two groups $A$ and $B$ such that the sum of the $x$-coordinates of all points in group $A$ is not greater than 6, and the sum of the $y$-coordinates of all points in group $B$ is not greater than 6, then what is the maximum value of the positive integer $k$?

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