
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
A valuable right-angled triangular metal plate $A B O$ is placed in a plane rectangular coordinate system (as shown in the diagram), with $A B = B O = 1$ (meter) and $A B \perp O B$. Due to damage in the shaded part of the triangular plate, a line $M N$ passing through the point $P\left(\frac{1}{2}, \frac{1}{4}\right)$ is needed to cut off the damaged part. How should the slope of the line $M N$ be determined such that the area of the resulting triangular plate $\triangle A M N$ is maximized?

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