
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 $f(x)=\ln x-\frac{1}{2}ax^2+x$, where $a\in \mathbb{R}$.

$(1)$ If $f(1)=0$, find the interval of monotonic decrease for the function $f(x)$.

$(2)$ If the inequality $f(x) \leqslant ax-1$ always holds for $x$, find the minimum value of the integer $a$.

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