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

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