
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 positive numbers $x$ and $y$ that satisfy $x + 2y = 3$, find the length of the chord of the tangent line to the circle $(x - \frac{1}{2})^2 + (y + \frac{1}{4})^2 = \frac{1}{2}$ passing through the point $P(x, y)$ when $xy$ reaches its maximum value.

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