
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 the function $f(x) = \frac{a}{3}x^3 - \frac{3}{2}x^2 + (a+1)x + 1$, where $a$ is a real number.
1. Given that the function $f(x)$ has an extremum at $x=1$, determine the value of $a$.
2. Given that the inequality $f'(x) > x^2 - x - a + 1$ holds for any $a \in (0, +\infty)$, find the range of real numbers $x$.

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