
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 such that the number in the $i$-th row and $j$-th column is denoted as $x_{i, j}$ for $i=1,2, \cdots, 100$ and $j=1,2, \cdots, 25$. After filling the table (referred to as Table 1), the numbers in each column are sorted in descending order to form a new table (referred to as Table 2), where the new values $x^{\prime}_{i, j}$ satisfy $x^{\prime}_{1, j} \geqslant x^{\prime}_{2, j} \geqslant \cdots \geqslant x^{\prime}_{100, j}$ for $j=1,2, \cdots, 25$.

Find the smallest natural number $k$ such that if the sums of the numbers in each row of Table 1 satisfy $\sum_{j=1}^{25} x_{i, j} \leqslant 1$ for $i=1,2, \cdots, 100$, then it is guaranteed that $\sum_{j=1}^{25} x^{\prime}_{i, j} \leqslant 1$ for all $i \geqslant k$ in Table 2.

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