
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
Determine all real numbers \( a, b, c, d \) that satisfy the following system of equations.
\[
\left\{\begin{array}{r}
a b c+a b+b c+c a+a+b+c=1 \\
b c d+b c+c d+d b+b+c+d=9 \\
c d a+c d+d a+a c+c+d+a=9 \\
d a b+d a+a b+b d+d+a+b=9
\end{array}\right.
\]

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