
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 a parabolic curve $y^{2}=2x$ with a point $P$ on it, points $B$ and $C$ are on the y-axis. A circle described by $(x-1)^{2}+y^{2}=1$ is inscribed within the triangle $PBC$. Find the minimum area of triangle $PBC$.

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