
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
Let the function $f(x)=(x-1)^{2}-a\ln x$, where $a\in\mathbb{R}$.  
$(1)$ If the tangent line to the curve $y=f(x)$ at the point $(1,f(1))$ is perpendicular to the line $x+2y-1=0$, find the value of $a$;  
$(2)$ Determine the intervals of monotonicity for the function $f(x)$.

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