
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 $f(x) = ax^2 + bx + c$ and $g(x) = dx + e$, and the equality $f(g(x)) = g(f(x))$ holds, find the necessary condition for $a$, $b$, $c$, $d$, and $e$.
**A)** $a(d - 1) = 0, \ ae = 0, \ c - e = ae^2$ \\
**B)** $ad = 1$ \\
**C)** $ae + b = ae^2 + be + c$ \\
**D)** $ad^2 = da, \ 2ae + b = b, \ c = e$ \\
**E)** None of the above

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