
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 $C: y^{2}=2px (p > 0)$, it circumscribes $\triangle ABC$ where $\angle A=90^{\circ}$.
- $(I)$ When point $A$ coincides with the origin, find the equation of the trajectory of the midpoint $M$ of the hypotenuse $BC$.
- $(II)$ When the ordinate of point $A$ is a constant $t_{0} (t_{0} \in \mathbb{R})$, determine whether the line containing $BC$ passes through a fixed point. If it does, find the coordinates of the fixed point; if not, explain why.

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