
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
Each vertex of a convex pentagon $ABCDE$ is to be assigned a color. There are $5$ colors to choose from, and the ends of each diagonal must have different colors. Additionally, vertices $A$ and $C$, as well as $B$ and $D$, must also not share the same color. How many different colorings are possible?
A) 480
B) 520
C) 540
D) 560
E) 600

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