
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
When the digits in the number $3014$ are reversed, we obtain the number $4103$. Assume $4103 = p \cdot q \cdot r$, where $p$, $q$, and $r$ are three distinct primes. How many other positive integers are the products of exactly three distinct primes $x$, $y$, and $z$ such that $x + y + z = p+q+r$?

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