
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 Cartesian coordinate system xOy, it is known that on the circle $C: (x - \sqrt{a})^2 + (y - a)^2 = 1$ ($a \geq 0$), there exists only one point P such that the distance from P to the line $l: y = 2x - 6$ equals $\sqrt{5} - 1$. Find the value of the real number $a$.

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