
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 $a\in R$, the functions are defined as follows: $f(x)=\frac{1}{3}{x^3}-\frac{1}{2}(a-1)x^{2}-ax-3$ and $g\left(x\right)=x-2\ln x$. Find:

1. When $a=1$, find the equation of the tangent line to the function $y=f\left(x\right)$ at the point $\left(3,f\left(3\right)\)$.
2. If the decreasing interval of the function $f\left(x\right)$ is $\left(-1,4\right)$, find the values of $a$.
3. If the function $y=g\left(x\right)-a$ has exactly two distinct zeros in the interval $\left[1,3\right]$, find the range of real numbers for $a$.

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