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What would be the recommended method to customize the quote environment such that every line starts with, say, opening curly quotes on the left margin.

Something like

enter image description here

\documentclass[12pt, oneside]{article}
\usepackage[font=small, noorphans, indentfirst=false, leftmargin=\parindent, rightmargin=0pt]{quoting}
\begin{document}
\begin{quoting}
One day Shizuo Kakutani was teaching a class at Yale. He wrote down a lemma on the blackboard and announced that the proof was obvious. One student timidly raised his hand and said that it wasn't obvious to him. Could Kakutani explain?

After several moments' thought, Kakutani realized that he could not himself prove the lemma. He apologized, and said that he would report back at their next class meeting.

After class, Kakutani, went straight to his office. He labored for quite a time and found that he could not prove the pesky lemma. He skipped lunch and went to the library to track down the lemma. After much work, he finally found the original paper. The lemma was stated clearly and succinctly. For the proof, the author had written, "Exercise for the reader."
\end{quoting}
\end{document}

(Obviously, I don't have to use the quoting package.)

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