
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
If $\dfrac{a+2b}{2b+c} = \dfrac{c+2d}{2d+a}$, then find the relationship between $a$, $b$, $c$, and $d$.
**A)** $a = c$   
**B)** $a+c+2(b+d) = 0$   
**C)** either $a = c$ or $a+c+2(b+d) = 0$, or both   
**D)** all variables individually equal   
**E)** no relationship exists

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