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I would like to have an object animate between Start_1 and End_1. Every 50 frames the ending value is given a new random value, and the preceding end value becomes the new start value.

In other words, where the last euler animation ended is where the new euler animation starts. This continues on indefinitely.

I imagine re-assigned a variable in a loop is involved, but I haven't been able to get much further than that.

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First, use the Repeat Time node to repeat the the animation every constant duration. Except, we are going to dynamically change the start and end of the animation. Now, if you floor divide the time by the duration, you will get an integer that is constant for the whole duration of the animation and increases by one every time the animation repeats. Since the random nodes are deterministic based on the seed, we can use that integer as the seed of a random node and use its output as the start of the animation, while the end of the animation will be a random node evaluated at the integer plus one as follows:

Node Tree

Note that the node seed is equal in both random nodes.

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