
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
Given real numbers $a, b, c, d$, consider matrix $\begin{pmatrix} a & b & c & d \\ b & c & d & a \\ c & d & a & b\\ d & a & b & c \end{pmatrix}$. Determine whether the matrix is invertible. If it is not, list all possible values of:
\[
\frac{a}{b + c + d} + \frac{b}{a + c + d} + \frac{c}{a + b + d} + \frac{d}{a + b + c}.
]

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