
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 $(1+0.3)^{800}$ by the binomial theorem and doing no further manipulation gives
\[{800 \choose 0}(0.3)^0+{800 \choose 1}(0.3)^1+{800 \choose 2}(0.3)^2+\cdots+{800 \choose 800}(0.3)^{800}= B_0 + B_1 + B_2 + \cdots + B_{800},\]where $B_k = {800 \choose k}(0.3)^k$ for $k = 0,1,2,\ldots,800.$ For which $k$ is $B_k$ the largest?

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