"Casts one net" can be replaced by the synonyms "tries one's luck," "takes a chance," "takes a risk," or "puts oneself out there." These phrases all imply the act of putting in effort or taking action in order to achieve a desired outcome, without any guarantee of success. Other synonyms for this expression may include "throws out a line," "launches a probe," "ventures forth," or "launches oneself," all of which capture the sense of exploring new territory or pursuing new opportunities. In any case, the underlying meaning of these phrases is one of initiative, determination, and a willingness to take a leap of faith in pursuit of one's goals.