
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
Let \( S \) be the set of ordered pairs of integers \( (x, y) \) with \( 1 \leq x \leq 5 \) and \( 1 \leq y \leq 3 \). How many subsets \( R \) of \( S \) have the property that all the points of \( R \) lie on the graph of a single cubic? A cubic is a polynomial of the form \( y = ax^3 + bx^2 + cx + d \), where \( a, b, c \), and \( d \) are real numbers (meaning that \( a \) is allowed to be 0).

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