
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 simplifying, we get
\[{500 \choose 0}(0.1)^0 + {500 \choose 1}(0.1)^1 + {500 \choose 2}(0.1)^2 + \cdots + {500 \choose 500}(0.1)^{500} = B_0 + B_1 + B_2 + \cdots + B_{500},\]
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}
user