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

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