
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 $f(x)=x+a\ln x$ at $x=1$, its tangent line is perpendicular to the line $x+2y=0$, and $g(x)=f(x)+ \frac {1}{2}x^{2}-bx$.
(1) Find the value of the real number $a$;
(2) Let $x_{1}$, $x_{2}(x_{1} < x_{2})$ be the two extreme points of the function $g(x)$, if $|g(x_{1})-g(x_{2})|\geqslant \frac {3}{4}-\ln 2$, find the range of $b$.

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