
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
Lunasa, Merlin, and Lyrica each has an instrument. We know the following about the prices of their instruments:

(a) If we raise the price of Lunasa's violin by  $50\%$  and decrease the price of Merlin's trumpet by  $50\%$ , the violin will be  $\$ 50 $ more expensive than the trumpet;
(b) If we raise the price of Merlin's trumpet by $ 50\% $ and decrease the price of Lyrica's piano by $ 50\% $, the trumpet will be $ \ $50$  more expensive than the piano.

Given these conditions only, there exist integers  $m$  and  $n$  such that if we raise the price of Lunasa's violin by  $m\%$  and decrease the price of Lyrica's piano by  $m\%$ , the violin must be exactly  $\$ n $ more expensive than the piano. Find $ 100m+n$.

*Proposed by Yannick Yao*

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