
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
Define $\max\{x_1, x_2, x_3, \ldots, x_n\}$ to be the maximum value among $x_1, x_2, x_3, \ldots, x_n$. Given the sequences $a_n=\frac{1000}{n}$, $b_n=\frac{2000}{m}$, $c_n=\frac{1500}{p}$, where $n+m+p=200$, $m=kn$, and $n$, $m$, $p$, $k$ are positive integers. Let $d_n=\max\{a_n, b_n, c_n\}$.
1. Find $\max\{a_n, b_n\}$.
2. When $k=2$, find the minimum value of $d_n$.
3. For every $k$ in positive integers, find the minimum value of $d_n$.

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