For certain real numbers \(a\), \(b\), and \(c\), the polynomial \[g(x) = x^3 + ax^2 + x + 10\] has three distinct roots, and each root of \(g(x)\) is also a root of the polynomial \[f(x) = x^4 + x^3 + bx^2 + 100x + c.\] What is the value of \(f(1)\)?

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