
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
The center of the circle $x^2 + y^2 - 2x - 4y + 1 = 0$ is the point $\left(1, 2\right)$. The line $ax + y - 1 = 0$ can be rewritten in the slope-intercept form as $y = ax + 1$. Determine the value of $a$ such that the distance from the point $\left(1, 2\right)$ to the line $y = ax + 1$ is equal to 1.

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