
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
Let vector $\overrightarrow{m} = (x, y)$, and vector $\overrightarrow{n} = (x-y)$. If P is a moving point on the curve $\overrightarrow{m} \cdot \overrightarrow{n} = 1$ (where $x > 0$), and the distance from point P to the line $x - y + 1 = 0$ is always greater than $\lambda$, then the maximum value of the real number $\lambda$ is ______.

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