
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) Find the x-coordinate of the point of tangency for the tangent line passing through the origin $O$.

(2) For all $x \in [1,+\infty)$, the inequality $f(x) \geq a(2x-x^2)$ always holds true. Find the range of values for the real number $a$.

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