
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
The numbers  $1,2,\ldots,49,50$  are written on the blackboard. Ann performs the following operation: she chooses three arbitrary numbers  $a,b,c$  from the board, replaces them by their sum  $a+b+c$  and writes  $(a+b)(b+c)(c+a)$  to her notebook. Ann performs such operations until only two numbers remain on the board (in total 24 operations). Then she calculates the sum of all  $24$  numbers written in the notebook. Let  $A$  and  $B$  be the maximum and the minimum possible sums that Ann san obtain.
Find the value of  $\frac{A}{B}$ .

*(I. Voronovich)*

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