
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
A trapezium is given with parallel bases having lengths  $1$  and  $4$ . Split it into two trapeziums by a cut, parallel to the bases, of length  $3$ . We now want to divide the two new trapeziums, always by means of cuts parallel to the bases, in  $m$  and  $n$  trapeziums, respectively, so that all the  $m + n$  trapezoids obtained have the same area. Determine the minimum possible value for  $m + n$  and the lengths of the cuts to be made to achieve this minimum value.

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