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I think it helps to think of energy as power to do something, rather than something tangible. It takes energy to break a bond, but energy is released when a bond is formed. That's because a bonded state is typically less energetic or "powerful" than an unbonded one, and therefore more stable. Matter is always looking for that stability. So, to break a bond means going against that natural attempt to gain stability, therefore requiring energy from outside.
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