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As per CIP sequence rules, the topmost $\ce{C}$ on the left gets 1st priority. Moving on to the right side of the double bond, we have:

Top $\ce{C}$: ($\ce{C}$,$\ce{O}$,$\ce{O}$,$\ce{H}$)
Bottom $\ce{C}$: ($\ce{C}$,$\ce{O}$,$\ce{O}$,$\ce{H}$,$\ce{C}$...)

Thus bottom $\ce{C}$ gets priority one on the right side.

Now, since 1st priorities are opposite, it's an E isomer. But the solution manual states that it's a Z isomer. Why is my method incorrect?

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CIP rules for cyclic substituents are different. See this question for the rules. (The rules are given in the question itself).

This will be the hierarchical digraph for the right side of the molecule.

enter image description here enter image description here

The atoms encircled are the ghost atoms. From the digraph one can see why the compound is Z.

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