
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 an \(m \times 10\) grid, each cell is filled with either a 0 or a 1 such that each column contains exactly three 1s. Let the sum of the numbers in the \(i^{th}\) row be denoted by \(x_i\). Furthermore, for any two columns, there is always at least one row where both columns have a 1. Let \(x = \max_{1 \leq i \leq m} \{x_i\}\). Find the minimum value of \(x\), denoted by \(x_{\text{min}} =\) ______.

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