
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
Consider a polynomial equation of the form $x^3 + bx^2 + cx + d = 0$, where the coefficients $b, c,$ and $d$ are selected from the set of integers $\{1, 2, 3, 4\}$. How many such equations have real roots?

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