
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 a function $f(x) = \ln x - mx^2$ and $g(x) = \frac{1}{2}mx^2 + x$, with $m \in \mathbb R$.

(Ⅰ) When $m = \frac{1}{2}$, find the interval where function $f(x)$ is monotonically increasing.

(Ⅱ) If the inequality $f(x) + g(x) \leq mx - 1$ holds true for all $x$, find the minimum integer value of $m$.

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