
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 $f(x) = ax^2 - (2a + 1)x + 2\ln x$ (where $a$ is a positive number),

(1) If the tangent lines to the curve $y = f(x)$ at $x = 1$ and $x = 3$ are parallel, find the value of $a$.

(2) Find the intervals of monotonicity for $f(x)$.

(3) Let $g(x) = x^2 - 2x$. If for any $x_1 \in (0, 2]$, there exists $x_2 \in (0, 2]$ such that $f(x_1) < g(x_2)$, determine the range of the real number $a$.

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