
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
Consider the following expansion using the binomial theorem: $(1+0.1)^{2000}$. This can be expanded as:
\[\sum_{k=0}^{2000} {2000 \choose k}(0.1)^k = B_0 + B_1 + B_2 + \cdots + B_{2000},\]
where $B_k = {2000 \choose k}(0.1)^k$. Identify the value of $k$ for which $B_k$ is the largest.

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