
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
(1) In the first quadrant of the Cartesian coordinate system, there are $(n+1)^{2}$ lattice points with coordinates $(i, j) (i, j \in \mathbf{Z}, 1 \leqslant i, j \leqslant n)$. Find the number of squares that can be formed using these lattice points as vertices (squares can be tilted).

(2) In space, is there a cube with vertices being lattice points such that none of its edges are parallel to the coordinate axes?

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