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1. A protestation is a strong declaration that something is true or not true.
A protest is the act of saying or showing publicly that you object to something.
只protest才符合第一句的上下文：to stage a protest = to organize a protest and take part in it＝進行抗議行動。

2. despite the rain = 不顧天下雨
如果寫成despite + V-ing時，動名詞V-ing在邏輯上的主詞是主句的主詞，如：
He was obvious distressed despite being unconscious.
being unconscious的主詞是he：即he was unconscious
如果是despite+一般名詞則無此問題，如：
She told friends she will stand by husband, despite reports that he sent another woman love notes.
即there were reports