
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
Express $y$ from the straight line equation $y-1=\frac{4-a x-b}{b}$ and substitute it into the circle equation:

$$b^{2}(x-1)^{2}+(a x+b-4)^{2}-b^{2}=0$$

Transform it to:

$$x^{2}\left(a^{2}+b^{2}\right)+2 x\left(a(b-4)-b^{2}\right)+(b-4)^{2}=0$$

Condition for the tangent line touching the circle is $D=0$, implying:

$$\left(a(b-4)-b^{2}\right)^{2}-\left(a^{2}+b^{2}\right)(b-4)^{2}=0$$

After transforming:

$$4 a+4 b-a b-8=0 \Rightarrow a=\frac{4 b-8}{b-4}=4+\frac{8}{b-4}$$

For $a$ to be an integer, $b-4$ must be a divisor of 8, i.e., $b-4=\pm 1, \pm 2, \pm 4, \pm 8$.

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