
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 $f(x) = \frac{1}{2}x^2 - a\ln{x}$ (where $a$ is a real number),
(1) If the tangent line of the function $f(x)$ at $x=2$ is given by $y=x+b$, find the values of $a$ and $b$;
(2) Discuss the number of solutions to the equation $f(x) = 0$, and explain the reasoning.

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