
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 circle \( \odot O: x^{2}+y^{2}=1 \) with a chord \( AB \) of length \( \sqrt{3} \) and a moving point \( P \) on the circle \( \odot C: (x-2)^{2}+(y-3)^{2}=2 \), find the minimum value of \( \sqrt{\overrightarrow{P A} \cdot \overrightarrow{P B}+\frac{3}{4}} \).

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