
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
Alice and Bob play a game. Alice writes an equation of the form  $ax^2 + bx + c =0$ , choosing  $a$ ,  $b$ ,  $c$  to be real numbers (possibly zero). Bob can choose to add (or subtract) any real number to each of  $a$ ,  $b$ ,  $c$ , resulting in a new equation. Bob wins if the resulting equation is quadratic and has two distinct real roots; Alice wins otherwise. For which choices of  $a$ ,  $b$ ,  $c$  does Alice win, no matter what Bob does?

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