
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
Suppose  $ n$  is a product of four distinct primes  $ a,b,c,d$  such that:
 $ (i)$   $ a\plus{}c\equal{}d;$ 
 $ (ii)$   $ a(a\plus{}b\plus{}c\plus{}d)\equal{}c(d\minus{}b);$ 
 $ (iii)$   $ 1\plus{}bc\plus{}d\equal{}bd$ .
Determine  $ n$ .

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