
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
Steve is piling \( m \geq 1 \) indistinguishable stones on the squares of an \( n \times n \) grid. Each square can have an arbitrarily high pile of stones. After he finished piling his stones in some manner, he can then perform stone moves, defined as follows: Consider any four grid squares, which are corners of a rectangle, i.e., in positions \( (i, k), (i, l), (j, k), (j, l) \) for some \( 1 \leq i, j, k, l \leq n \), such that \( i < j \) and \( k < l \). A stone move consists of either:
1. Removing one stone from each of \( (i, k) \) and \( (j, l) \) and moving them to \( (i, l) \) and \( (j, k) \) respectively, or
2. Removing one stone from each of \( (i, l) \) and \( (j, k) \) and moving them to \( (i, k) \) and \( (j, l) \) respectively.

Two ways of piling the stones are equivalent if they can be obtained from one another by a sequence of stone moves. How many different non-equivalent ways can Steve pile the stones on the grid?

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