
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 \( (1+0.1)^{500} \) using the binomial theorem results in:
\[ \sum_{k=0}^{500} \binom{500}{k} (0.1)^k = A_0 + A_1 + A_2 + \cdots + A_{500}, \]
where \( A_k = \binom{500}{k}(0.1)^k \) for \( k = 0, 1, 2, \ldots, 500 \). For which \( k \) is \( A_k \) the largest?

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