
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 row $i$ be denoted as $x_i$ for $i = 1, 2, \ldots, m$. Furthermore, for any two columns, there always exists at least one row where the cells at the intersection of this row and these two columns are both filled with 1s. Let $x = \max_{1 \leq i \leq m} \{ x_i \}$. What is the minimum value of $x$?

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