
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 points $A(a-1, a)$ and $B(a, a+1)$, and a moving point $P$ such that its distance to the point $M(1,0)$ is greater by 1 than its distance to the $y$-axis. The locus of $P$ is a curve $C$. The line segment $AB$ intersects curve $C$ at exactly one point. Determine the range of the value of $a$.

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