
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 functions $f(x) = \frac{1}{2}mx^2 + 1$, $g(x) = 2\ln x - (2m+1)x - 1$ ($m \in \mathbb{R}$), and $h(x) = f(x) + g(x)$,  
(1) If the tangents of the function $h(x)$ at $(1, f(1))$ and $(3, f(3))$ are parallel, find the value of the real number $m$;  
(2) Determine the intervals of monotonicity for $h(x)$.

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