
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
A set  $A$  is endowed with a binary operation  $*$  satisfying the following four conditions:
(1) If  $a, b, c$  are elements of  $A$ , then  $a * (b * c) = (a * b) * c$  ,
(2) If  $a, b, c$  are elements of  $A$  such that  $a * c = b *c$ , then  $a = b$  ,
(3) There exists an element  $e$  of  $A$  such that  $a * e = a$  for all  $a$  in  $A$ , and
(4) If a and b are distinct elements of  $A-\{e\}$ , then  $a^3 * b = b^3 * a^2$ , where  $x^k = x * x^{k-1}$  for all integers  $k \ge 2$  and all  $x$  in  $A$ .
Determine the largest cardinality  $A$  may have.

proposed by Bojan Basic, Serbia

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