
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
Determine, with proof, the number of ordered triples  $\left(A_{1}, A_{2}, A_{3}\right)$  of sets which have the property that

(i)  $A_{1} \cup A_{2} \cup A_{3}=\{1,2,3,4,5,6,7,8,9,10\},$  and
(ii)  $A_{1} \cap A_{2} \cap A_{3}=\emptyset.$ Express your answer in the form  $2^{a} 3^{b} 5^{c} 7^{d},$  where  $a, b, c, d$  are nonnegative integers.

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