
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

The game takes place on a $9 \times 9$ grid of squared paper. Two players take turns; the player who starts the game places crosses in free cells while their partner places noughts. When all the cells are filled, the number of rows and columns $K$ in which crosses outnumber noughts, and the number of rows and columns $H$ in which noughts outnumber crosses, is counted. The difference $B=K-H$ is considered the winning score of the player who starts. Find the value of $B$ such that:

1) The first player can ensure a win of at least $B$, regardless of how the second player plays;
2) The second player can always ensure that the first player’s winning score is no more than $B$, regardless of how the first player plays.

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