
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 the ratio of the distances from a point to (1, 0) and to (4, 0) is 1:2, and there are exactly 3 points at a distance of 1 from the line $y = x + c$, then the value of $c$ is $\_\_\_\_\_$.

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