
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 \( n \) expressions of the form \({ }^{*} \chi^{2} + *_{X} + * = 0\) written on the board (\( n \) is an odd number). Two players play a game where they take turns. On each turn, one is allowed to replace one of the asterisks with a non-zero number. After \( 3n \) moves, \( n \) quadratic equations are completed. The first player aims to maximize the number of these equations that have no roots, while the second player tries to prevent this. What is the maximum number of equations without roots that the first player can achieve regardless of the second player's strategy?

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