
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
For every sequence  $p_1<p_2<\cdots<p_8$  of eight prime numbers, determine the largest integer  $N$  for which the following equation has no solution in positive integers  $x_1,\ldots,x_8$ :  $$ p_1\, p_2\, \cdots\, p_8 
\left( \frac{x_1}{p_1}+ \frac{x_2}{p_2}+ ~\cdots~ +\frac{x_8}{p_8} \right)
~~=~~ N  $$  *Proposed by Gerhard Woeginger, Austria*

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