
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
Determine the value of $b$ for which there is a smallest positive real number $a$ such that for this value of $a$, a polynomial $x^3 - ax^2 + bx - b = 0$ has all its roots real and positive, where $b=a^2/3$.

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