
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
For the function $f(x) = x^2 + 2x$, among all the constants $M$ such that $f(x) \geq M$ is satisfied, we call the greatest value of $M$ the greatest lower bound (infimum) of $f(x) = x^2 + 2x$. Then, the greatest lower bound (infimum) for $ax^2 + bx$ where $a, b \in \mathbb{R}$ and not both $a$ and $b$ are zero, is ( ).

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