
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 $f(x) = \frac{1}{2}ax^2 - (2a+1)x + 2\ln x$ ($a \in \mathbb{R}$),  
(1) When $a = \frac{2}{3}$, find the intervals of monotonicity for the function $f(x)$.  
(2) Let $g(x) = (x^2 - 2x)e^x$. If for any $x_1 \in (0, 2]$, there exists $x_2 \in (0, 2]$ such that $f(x_1) < g(x_2)$ holds, find the range of the real number $a$.

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