
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 $f(x)=\ln x+ax^{2}+bx$ where $(a,b\in R)$:
1. If $a=0$ and $b=1$, find the equation of the tangent line to the function $y=f(x)$ at the point $P(1,f(1))$.
2. If $a=\frac{1}{2}$ and $b=-3$, point $Q$ is a moving point on the function $y=f(x)$. Let $\alpha$ be the angle of the tangent line at point $Q$. Find the range of values for the angle $\alpha$.
3. If $b=-1$, for any $x_{1}$ and $x_{2}\in (0,+\infty)$ where $x_{1}\neq x_{2}$, it is given that $\frac{f(x_1)-f(x_2)}{x_1-x_2} > 1$. Find the range of real numbers for $a$.

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