
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 $f(x) = x^2 - a\ln{x}$ $(a\in\mathbb{R})$,  
(1) If $a=2$, find the extremum of the function $f(x)$.  
(2) Given that the tangent line $l$ at point $A(1, f(1))$ passes through the graph of $y=f(x)$ at point $A$ (meaning that as the moving point $P$ on the graph of $y=f(x)$ passes through point $A$, it moves from one side of $l$ to the other), find the expression for the function $f(x)$.  
(3) If $a > 0$ and the function $g(x) = f(x) - ax$ has exactly one zero, find the value of the real number $a$.

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