What tense is the phrase "is thought to have been" in the sentence "Bruce Lee is thought to have been the first actor to do his own stunts"?
Also, why is it correct to say "to do his own stunts" instead of "to have done"?
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For the purpose of determining the tense, the is is all that matters. The phrase is in Present Simple.
In fact, depending on your definition of tense (which does vary wildly between learners of the language and linguists), English doesn't have terribly many to begin with.
As to "to do his own stunts" vs. "to have done", both are equally grammatical. And in this particular context, I'd even add that they don't bring to the table a noteworthy difference in meaning.