
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
In the right triangular prism \( ABC-A_1B_1C_1 \), \( AB=1 \), \( BC=C_1C=\sqrt{3} \), and \( \angle ABC=90^\circ \). Point \( P \) is a moving point on plane \( ABC \). Find the minimum value of \( A_1P + \frac{1}{2}PC \).

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