The given reaction is an example of the Hell-Volhard-Zelinsky (HVZ) reaction, which selectively brominates the $\alpha$-carbon of carboxylic acids in the presence of Br$_2$ and red phosphorus. The reaction mechanism involves the formation of an $\alpha$-brominated carboxylic acid. \[\text{COOH} \xrightarrow{\text{Br}_2/\text{Red P}} \text{COOH-Br} \xrightarrow{\text{H}_2\text{O}} \text{COOH-Br (Product)}\] Thus, the product formed is the $\alpha$-bromo derivative of the given carboxylic acid.
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