
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 $f(x)=ax-\ln x$.  
$(1)$ Draw a tangent line from the origin $O$ to the curve $y=f(x)$, and find the $x$-coordinate of the tangent point;  
$(2)$ For $\forall x \in [1, +\infty)$, the inequality $f(x) \geqslant a(2x-x^{2})$ holds, find the range of the real number $a$.

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