Nineteen weights of mass $1$ gm, $2$ gm, $3$ gm, \ldots, $19$ gm are given. Nine are made of iron, nine are of bronze, and one is pure gold. It is known that the total mass of all the iron weights is $90$ gm more than the total mass of all the bronze ones. Find the mass of the gold weight.

Let's think step by step and output the final answer within \boxed{}.
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