
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 parabola $C_1: y = \frac{1}{2p}x^2$ with $p > 0$ and a hyperbola $C_2: \frac{x^2}{8}-y^2=1$, find the value of $p$ such that the focus of $C_1$ is connected to the right focus of $C_2$ by a line intersecting $C_1$ at the first quadrant point $M$, and the tangent line of $C_1$ at point $M$ is parallel to one asymptote of $C_2$.

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