
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
If a real number $x_{0}$ satisfies $f(x_{0})=x_{0}$, then $x=x_{0}$ is called a fixed point of $f(x)$. Given the function $f(x)=x^{3}+bx+3$, where $b$ is a constant.  
(Ⅰ) Find the interval of monotonic increase for the function $f(x)$.  
(Ⅱ) If there exists a real number $x_{0}$, such that $x=x_{0}$ is both a fixed point and an extremum of $f(x)$. Find the value of the real number $b$.

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