Let's count the opening parentheses in the given string:
"()((()(()((()()((()))()((()(()(((()((()())))()(())()))())()(())((()()((()(()((()()((()))()((()(()(((()((()())))()(())()))())()(())(((()((()())))()(())()))())()(()))()(((()((()((()()((()))()((()(()(((()((()())))()(())()))())()(())(((()((()())))()(())()))())()(()))()))())()(())(((()((()())))()(())()))())()(()"

We can manually count the opening parentheses, "(" as follows:
1: (
2: (
3: (
4: (
5: (
6: (
7: (
8: (
9: (
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11: (
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157: (
158: (
159: (
160: (

There are 160 opening parentheses.

Final Answer: The final answer is $\boxed{160}$
