
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
A $100 \times 25$ rectangular array is filled with non-negative real numbers, where the number in the $i^{th}$ row and $j^{th}$ column is denoted by $x_{ij}$ for $i=1, 2, \cdots, 100$ and $j=1, 2, \cdots, 25$. The numbers in each column are then rearranged in non-increasing order to form a new array:
$$
x'_{1j} \geq x'_{2j} \geq \cdots \geq x'_{100j}, \quad j=1, 2, \cdots, 25
$$
Find the smallest natural number $k$ such that if the numbers in the original array satisfy
$$
\sum_{j=1}^{25} x_{ij} \leq 1 \quad \text{for} \; i=1, 2, \cdots, 100,
$$
then for $i \geq k$, it holds that
$$
\sum_{j=1}^{25} x'_{ij} \leq 1
$$
in the new array.

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