
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 the real numbers \(a\) and \(b\) such that the equation \(a x^{3} - x^{2} + b x - 1 = 0\) has three positive real roots. For all real numbers \(a\) and \(b\) that meet the conditions, find the minimum value of \(P = \frac{5 a^{2} - 3 a b + 2}{a^{2}(b - a)}\).

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