
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 $f(x) = x^2 - (2a+1)x + a\ln(x)$, where $a \in \mathbb{R}$,  
(1) If the tangent line to the function $f(x)$ at the point $(1, f(1))$ is perpendicular to the $y$-axis, find the value of the real number $a$;  
(2) Discuss the intervals of monotonicity for the function $f(x)$;  
(3) If for $x > 1$, $f(x) > 0$ always holds, determine the range of values for the real number $a$.

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