
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
In the rectangular coordinate system $xOy$, there are parabolas $C_{i}: y=a_{i} x^{2}+b_{i} x+c_{i} (i=1, 2, \cdots, 20)$ and parabolas $D_{j}: x=a_{j}^{\prime} y^{2}+b_{j}^{\prime} y+c_{j}^{\prime} (j=1, 2, \cdots, 20)$. For any real numbers $a_{i}, b_{i}, c_{i}, a_{j}^{\prime}, b_{j}^{\prime}, c_{j}^{\prime}$, what is the maximum number of regions into which these 40 parabolas divide the coordinate plane?

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