
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 \((1+0.1)^{500}\) by the binomial theorem and doing no further manipulation gives
\[
{500 \choose 0}(0.1)^0+{500 \choose 1}(0.1)^1+{500 \choose 2}(0.1)^2+\cdots+{500 \choose 500}(0.1)^{500}= B_0 + B_1 + B_2 + \cdots + B_{500},
\]
where \(B_k = {500 \choose k}(0.1)^k\) for \(k = 0,1,2,\ldots,500.\) For which \(k\) is \(B_k\) the largest?

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