
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 yields 
\[ \sum_{k=0}^{2000} \binom{2000}{k} (0.1)^k = A_0 + A_1 + \dots + A_{2000}, \]
where $A_k = \binom{2000}{k} (0.1)^k$ for $k = 0, 1, 2, \ldots, 2000$. Determine the value of $k$ for which $A_k$ is the largest.

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