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Pour a kettle of boiling water into a very large pan until half full. Return to the boil, then add the potatoes and spaghetti, and a little salt. Cook for 10 minutes until the potatoes and pasta are almost tender. Tip in the green beans and cook for 5 minutes more.
Drain well, reserving 4 tbsp of the cooking liquid. Return the potatoes, pasta and beans to the pan, then stir in the fresh pesto and reserved cooking liquid. Season to taste, divide between four serving plates and drizzle with a little olive oil.