
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 functions $f(x)=x^{2}\ln x+ax$ and $g(x)=-x^{2}+bx-3$,

    (1) If the tangent line of the function $f(x)$ at the point $(1,f(1))$ is perpendicular to the line $2x+y-1=0$, find the value of the real number $a$;

    (2) When $a=0$, if the equation $xg(x)=2f(x)$ has two distinct real roots in the interval $\left( \frac{1}{2},2\right)$, find the range of values for the real number $b$ (given $\ln 2=0.69$).

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