
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 by $x_{i, j}$ $(i=1,2,\ldots, 100; j=1,2,\ldots, 25)$ (Table 1). The numbers in each column of Table 1 are then reordered in descending order to create Table 2 such that $x_{1, j}^{\prime} \geq x_{2, j}^{\prime} \geq \cdots \geq x_{100, j}^{\prime}$ $(j=1,2,\ldots, 25)$. Find the smallest natural number $k$ such that if the numbers in Table 1 satisfy $\sum_{j=1}^{25} x_{i, j} \leq 1$ $(i=1,2,\ldots, 100)$, then for $i \geq k$, Table 2 satisfies $\sum_{j=1}^{25} x_{i, j}^{\prime} \leq 1$ $(i=1,2,\ldots, 100)$.

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