
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 parabola $\Gamma$: $y^{2}=2px(p > 0)$ with focus $F$. If a line passing through point $F$ with a slope of $1$ intersects the parabola $\Gamma$ at points $M$ and $N$, and the area of $\triangle MON$ is $S_{\triangle MON}= \frac { \sqrt {2}}{2}$.  
$(1)$ Find the equation of the parabola $\Gamma$;  
$(2)$ If point $P$ is a moving point on the parabola $\Gamma$, and points $B$, $C$ are on the $y$-axis, and the circle $(x-1)^{2}+y^{2}=1$ is tangent to $\triangle PBC$, find the minimum area of $\triangle PBC$.

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