
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 that $a > 0$, the function is defined as $f(x)=\frac{x^{2}}{2}+2a(a+1)\ln x-(3a+1)x$.

(1) If the tangent line of the function $f(x)$ at $x=1$ is parallel to the line $y-3x=0$, find the value of $a$.

(2) Find the interval where the function $f(x)$ is strictly increasing.

(3) Under the condition of (1), if $f(x)-b^{2}-6b\geqslant 0$ holds for any $x\in [ 1,2 ]$, find the set of values of the real number $b$.

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