
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
The condition for the function \( y = \sqrt{x - m} \) and its inverse function to share common points on their graphs is that the graph of the function must be symmetric with respect to the line \( y = x \). This means the equation \( x = \sqrt{y - m} \) must be equivalent to the original equation, and the domain of the function must be non-negative. By equating the two equations, we can find the value of \( m \). The given function equation can be rewritten as \( y^2 = x - m \), and when substituting for \( x \) as \( y \), the equation becomes \( y^2 - m = y \). Rearranging this equation gives us the quadratic equation \( y^2 - y - m = 0 \). We want this equation to have a discriminant of zero, which would mean the parabola is tangential to the line y = x, giving the equation 1 - 4m = 0.

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