
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, every cell is filled with a non-negative real number, with the number in the $i$-th row and $j$-th column denoted as \( x_{ij} \), referred to as Table 1. Next, the numbers in each column of Table 1 are rearranged in descending order from top to bottom to form \( x'_{1j} \geq x'_{2j} \geq \cdots \geq x'_{100j} \) for \( j=1,2, \cdots, 25 \); this is referred to as Table 2.

Find the minimum positive integer \( k \) such that if the numbers in Table 1 satisfy \(\sum_{j=1}^{25} x_{ij} \leq 1\) for \(i=1,2,\cdots,100\), then for \(i \geq k\), the following condition in Table 2 is guaranteed: \(\sum_{j=1}^{25} x'_{ij} \leq 1\).

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