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: Expanding $(1+0.2)^{1000}$ by the binomial theorem and doing no further manipulation gives
\[{1000 \choose 0}(0.2)^0+{1000 \choose 1}(0.2)^1+{1000 \choose 2}(0.2)^2+\cdots+{1000 \choose 1000}(0.2)^{1000}= A_0 + A_1 + A_2 + \cdots + A_{1000},\]where $A_k = {1000 \choose k}(0.2)^k$ for $k = 0,1,2,\ldots,1000.$ For which $k$ is $A_k$ the largest?
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