
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 $f(x)=\ln x- \frac{1}{2}x^{2}$, $g(x)= \frac{1-m}{2}x^{2}+x$, where $m \in \mathbb{R}$, let $F(x)=f(x)+g(x)$.

(1) Find the interval of monotonic increase for the function $f(x)$.

(2) If the inequality $F(x) \leqslant mx-1$ always holds for $x$, find the minimum integer value of $m$.

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