
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 ellipse $mx^{2}+ny^{2}=1\ (m > 0,n > 0)$ intersects with the line $y=1-x$ at points $A$ and $B$. The slope of the line passing through the origin and the midpoint of the line segment $AB$ is $\frac{\sqrt{2}}{2}$. Find the value of $\frac{n}{m}$.

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