
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 circle $C: x^{2}+y^{2}-2x-2y+1=0$, and a line $l: y=kx$, the line $l$ intersects the circle $C$ at points $A$ and $B$. The coordinates of point $M$ are $(0,b)$, and it satisfies $\overrightarrow{MA} \perpendicular \overrightarrow{MB}$.

$(1)$ When $b=1$, find the value of $k$;

$(2)$ When $b \in \left( 1, \frac{3}{2} \right)$, find the range of values for $k$.

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