Solve the following math problem efficiently and clearly:

- For simple problems (2 steps or fewer):
Provide a concise solution with minimal explanation.

- For complex problems (3 steps or more):
Use this step-by-step format:

## Step 1: [Concise description]
[Brief explanation and calculations]

## Step 2: [Concise description]
[Brief explanation and calculations]

...

Regardless of the approach, always conclude with:

Therefore, the final answer is: $oxed{answer}$. I hope it is correct.

Where [answer] is just the final number or expression that solves the problem.

Problem: For any positive integer $n,$ let $\langle n \rangle$ denote the closest integer to $\sqrt{n}.$  Evaluate
\[\sum_{n = 1}^\infty \frac{2^{\langle n \rangle} + 2^{-\langle n \rangle}}{2^n}.\]
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