
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
Using the binomial theorem, expand \((1+0.3)^{1500}\) into
\[\binom{1500}{0}(0.3)^0 + \binom{1500}{1}(0.3)^1 + \cdots + \binom{1500}{1500}(0.3)^{1500} = A_0 + A_1 + \cdots + A_{1500}, \]
where \(A_k = \binom{1500}{k}(0.3)^k\) for \(k = 0, 1, 2, \ldots, 1500\). Determine which $k$ maximizes $A_k$.

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