
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 $f(x)=x^{3}+ax^{2}-3x (a\in R)$.
(1) Find the range of values of the real number $a$ such that the function $f(x)$ is increasing in the interval $[1,+\infty)$.
(2) If $x= \frac {1}{3}$ is the extreme point of the function $f(x)$, find the maximum value of the function $f(x)$ in the interval $[-a,1]$.
(3) Under the condition of (2), does there exist a real number $b$ such that the graph of the function $g(x)=bx$ intersects with the graph of the function $f(x)$ at exactly 3 points? If it exists, find the range of values of $b$; if not, explain the reason.

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