
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 $10 \times 10$ grid, fill in the integers $1, 2, \ldots, 100$ such that the sum of the numbers in every two adjacent cells (cells that share a common edge are called adjacent cells) is not greater than $S$. Find the minimum value of $S$.

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