
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
There are five "equations" of the form \( x^{2} + \ldots x + \ldots = 0 \) written on the board. Two players take turns filling in the blanks with natural numbers from 1 to 10, with each number being used exactly once. The game ends when all numbers are filled in. The player who makes the first move aims to maximize the number of equations that have two distinct roots at the end of the game, while their opponent aims to minimize this number. What is the best possible outcome that the first player can achieve regardless of the second player's moves?

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