
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 a rectangular parallelepiped $\mathrm{ABCDA}_{1} \mathrm{~B}_{1} \mathrm{C}_{1} \mathrm{D}_{1}$, with $\mathrm{AB}=1$ cm, $\mathrm{AD}=2$, and $\mathrm{AA}_{1}=1$, find the minimum area of triangle $\mathrm{PA}_{1} \mathrm{C}$, where vertex $\mathrm{P}$ lies on the line $\mathrm{AB}_{1}$.

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