
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
Expanding $(1+0.1)^{2000}$ by the binomial theorem and doing no further manipulation gives
\[{2000 \choose 0}(0.1)^0+{2000 \choose 1}(0.1)^1+{2000 \choose 2}(0.1)^2+\cdots+{2000 \choose 2000}(0.1)^{2000}= B_0 + B_1 + B_2 + \cdots + B_{2000},\]where $B_k = {2000 \choose k}(0.1)^k$ for $k = 0,1,2,\ldots,2000.$ For which $k$ is $B_k$ the largest?

Present the answer in LaTex format: \boxed{Your answer}
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