
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 plane Cartesian coordinate system $xOy$, $P$ is a moving point on the right branch of the hyperbola $x^2 - y^2 = 1$. If the distance from point $P$ to the line $x - y + 1 = 0$ is always greater than $m$, then the maximum value of the real number $m$ is $\boxed{\frac{\sqrt{2}}{2}}$.

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