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You shouldn't rely on other people too much.
Literal
Too-much others [on-に] rely-[then-て]-is-not-allowed.
A compact example of the あまり~ない pattern (here あまり~てはいけない) meaning 'don't do X excessively.' 頼る takes an に-marked object — 'to rely on X.' The construction ~てはいけない ('must not') is the standard prohibition form; dropping the は (~ちゃいけない) yields the casual version.