
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 grid, each cell contains a non-negative real number. The number in the $i$-th row and $j$-th column is denoted as $x_{i, j}$ $(i = 1, 2, \ldots, 100; j = 1, 2, \ldots, 25)$. Then, the numbers in each column are rearranged in descending order from top to bottom to form a new grid with entries $x^{\prime}_{i, j}$ such that $x^{\prime}_{1, j} \geq x^{\prime}_{2, j} \geq \cdots \geq x^{\prime}_{100, j}$ for $j = 1, 2, \ldots, 25$. 

Find the smallest natural number $k$ such that if the entries in the original grid satisfy $\sum_{j=1}^{25} x_{i, j} \leq 1$ for all $i = 1, 2, \ldots, 100$, then for $i \geq k$, it holds that $\sum_{j=1}^{25} x^{\prime}_{i, j} \leq 1$ in the rearranged grid.

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