
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
On a blackboard, there are $n$ (where $n$ is an odd number) expressions of the form $* x^{2}+* x+*=0$. Two players, A and B, take turns in the following game: each person can replace one of the * symbols with a non-zero real number. After $3n$ turns, the expressions become $n$ quadratic equations. Player A tries to maximize the number of equations without real roots, while Player B tries to minimize this number. What is the maximum number of equations without real roots that Player A can guarantee, regardless of Player B's actions?

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