
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
Let $c$ be a constant chosen from $[-10, 10]$ and let $b$ be a real number randomly selected from the interval $[-23, 23]$. Then, find and simplify the expression for the probability that the equation $x^3 + (c-2)b^2 = (2b^2 - 5cb)x$ has at least one real non-zero solution for $x$. Express the result as a simplified fraction $m/n$ and find $m+n$.

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