
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
In the plane rectangular coordinate system $xOy$, a line $l$ with a slope of $135^{\circ}$ passes through the point $\left(0,b\right)$ and intersects the circle $x^{2}+y^{2}-4x=0$ at points $A$ and $B$. If $|AB|=2\sqrt{2}$, calculate the value of $b$.

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