The third person singular form of rely is "relies."
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If "lay" is the infinite or the third person singular present indicative form, its past participle is "laid". Please note, however, that "lay" is also the past indicative form of the very confusingly similar verb "to lie", when that means to rely on something for support of one's body. When "to like" means "to falsify", its past participle is regular, "lied"!
The past participle form of "rely" is "relied."
The noun forms for the verb 'rely' are reliability and reliance.
The preposition that typically follows the verb "rely" is "on." For example, you would say "I rely on my friends for support." This preposition indicates the object or person on which the action of relying is directed.
Relying is a present participle. It is the -ing form of the verb "rely."