
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
Mary mixes  $2$  gallons of a solution that is  $40$  percent alcohol with  $3$  gallons of a solution that is  $60$  percent alcohol. Sandra mixes  $4$  gallons of a solution that is  $30$  percent alcohol with  $\frac{m}{n}$  gallons of a solution that is  $80$  percent alcohol, where  $m$  and  $n$  are relatively prime positive integers. Mary and Sandra end up with solutions that are the same percent alcohol. Find  $m + n$ .

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