
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

In a $100 \times 25$ rectangular table, each cell is filled with a non-negative real number. The number in the $i$-th row and $j$-th column is denoted as $x_{i,j}$ (where $i = 1, 2, \cdots, 100$ and $j = 1, 2, \cdots, 25$). The numbers in each column of this table are then rearranged in descending order from top to bottom as follows:
$$
x^{\prime}_{1,j} \geq x^{\prime}_{2,j} \geq \cdots \geq x^{\prime}_{100,j} \quad (j = 1, 2, \cdots, 25).
$$

Find the smallest natural number $k$ such that if the following condition is satisfied for all rows in the first table:
$$
\sum_{j=1}^{25} x_{i, j} \leq 1 \quad (i = 1, 2, \cdots, 100),
$$

then for all rows $i \geq k$ in the second table, the following inequality holds:
$$
\sum_{j=1}^{25} x^{\prime}_{i, j} \leq 1.
$$

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