
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 an integer  $n > 1$  be given. In the space with orthogonal coordinate system  $Oxyz$  we denote by  $T$   the set of all points  $(x, y, z)$  with  $x, y, z$  are integers, satisfying the condition:  $1 \leq x, y, z \leq n$ . We  paint all the points of  $T$  in such a way that: if the point  $A(x_0, y_0, z_0)$  is painted then points  $B(x_1, y_1, z_1)$  for which  $x_1 \leq x_0, y_1 \leq y_0$  and  $z_1 \leq z_0$  could not be painted. Find the maximal number of points that we can paint in such a way the above mentioned condition is satisfied.

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