
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 cell at the $i$-th row and $j$-th column is $x_{i,j}$, where $i=1, 2, \cdots, 100$ and $j=1, 2, \cdots, 25$. The table is represented as Table 1. The values in each column of Table 1 are then sorted in descending order from top to bottom to form Table 2, with the values in the $j$-th column being $x'_{1,j} \geq x'_{2,j} \geq \cdots \geq x'_{100,j}$, where $j=1, 2, \cdots, 25$.

Find the smallest natural number $K$ such that if the values in Table 1 satisfy $\sum_{j=1}^{25} x_{i,j} \leq 1$ for all $i=1, 2, \cdots, 100$, then for $i \geq K$, it is guaranteed that $\sum_{j=1}^{25} x'_{i,j} \leq 1$ in Table 2.

Table 1:
\[
\begin{array}{|c|c|c|c|}
\hline
x_{1,1} & x_{1,2} & \cdots & x_{1,25} \\
\hline
x_{2,1} & x_{2,2} & \cdots & x_{2,25} \\
\hline
\cdots & \cdots & \cdots & \cdots \\
\hline
x_{100,1} & x_{100,2} & \cdots & x_{100,25} \\
\hline
\end{array}
\]

Table 2:
\[
\begin{array}{|c|c|c|c|}
\hline
x'_{1,1} & x'_{1,2} & \cdots & x'_{1,25} \\
\hline
x'_{2,1} & x'_{2,2} & \cdots & x'_{2,25} \\
\hline
\cdots & \cdots & \cdots & \cdots \\
\hline
x'_{100,1} & x'_{100,2} & \cdots & x'_{100,25} \\
\hline
\end{array}
\]

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