
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 ellipse $C_1$ and parabola $C_2$ whose foci are both on the x-axis, the center of $C_1$ and the vertex of $C_2$ are both at the origin $O$. Two points are taken from each curve, and their coordinates are recorded in the table. Calculate the distance between the left focus of $C_1$ and the directrix of $C_2$.

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