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In a large pot, cook the pasta according to the package directions, but remove the pasta when al dente, about 1 minute early. Reserve 1/2 cup of the cooking water. Drain the pasta, return it to the pot, and toss with a little of the oil to prevent sticking.
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In a large skillet over medium-high heat, warm the 3 tablespoons oil. Add the onion, 1/2 teaspoon salt, and the red pepper flakes and sauté, stirring occasionally, until the onion softens, 4 to 5 minutes. Add the zucchini, sauté for 3 minutes longer, then add the tomatoes, another 1/2 teaspoon salt, and season with black pepper. Cook, stirring occasionally, until the tomatoes soften and zucchini is tender, 3 to 4 minutes.
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Add the spinach and the reserved cooking water to the pasta. Pour the vegetable mixture over the pasta, toss, and warm over medium heat for 2 minutes. Add the vinegar and toss. Add the olives and half of the feta and gently toss to combine.
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Divide the pasta among dinner plates, drizzle with extra-virgin olive oil, top with the remaining feta, and serve.