
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
Given the function $f(x) = \frac{1}{3}x^3 + \frac{1}{2}(2a+1)x^2 - 2(a+1)x$.

1. If $f(x)$ has a local maximum at $x = 1$, find the range of the real number $a$.
2. If there exists $x \in [1, 2]$ such that $f(x) \leq 0$, find the range of the real number $a$.

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