Sarah the Squirrel's Square Search

Sarah the squirrel is trying to find her acorn, but she can't remember where she left it! She starts in the lower-left corner of the \(2\times 2\) grid, and at each point, she randomly steps to one of the adjacent vertices (so she may accidentally travel along the same edge multiple times). What is the expected value for the number of steps Sarah will take before she finds her acorn in the top-right corner?

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