
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
There are 1988 identical unit cubes. Using all or some of these cubes, form three "squares" $A$, $B$, and $C$ with side lengths $a$, $b$, and $c$ respectively $(a \leq b \leq c)$ such that $A$, $B$, and $C$ are $a \times a \times 1$, $b \times b \times 1$, and $c \times c \times 1$ one-layer rectangular blocks.

First, place square $C$ on the plane. Then place $B$ on top of $C$ such that each small unit of $B$ is exactly above a small unit of $C$, but none of the outer edges of $B$ align with the edges of $C$. Finally, place $A$ on top of $B$ following the same principle, thus forming a three-layer structure.

What values of $a$, $b$, and $c$ maximize the number of different possible "three-layer structures"?

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