
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

Solve the systems of equations:

a) 
\[
\begin{cases}
x_{1} + x_{2} + x_{3} = 0 \\
x_{2} + x_{3} + x_{4} = 0 \\
\vdots \\
x_{99} + x_{100} + x_{1} = 0 \\
x_{100} + x_{1} + x_{2} = 0
\end{cases}
\]

b)
\[
\begin{cases}
x + y + z = a \\
y + z + t = b \\
y + z + t = c \\
t + x + y = d
\end{cases}
\]

c)
\[
\begin{cases}
x_{1} + x_{2} + x_{3} + x_{4} = 2a_{1} \\
x_{1} + x_{2} - x_{3} - x_{4} = 2a_{2} \\
x_{1} - x_{2} + x_{3} - x_{4} = 2a_{3} \\
x_{1} - x_{2} - x_{3} + x_{4} = 2a_{4}
\end{cases}
\]

d)
\[
\begin{cases}
x_{1} + 2x_{2} + 3x_{3} + \ldots + nx_{n} = a_{1} \\
nx_{1} + x_{2} + 2x_{3} + \ldots + (n-1)nx_{n} = a_{2} \\
\vdots \\
2x_{1} + 3x_{2} + 4x_{3} + \ldots + x_{n} = a_{n}
\end{cases}
\]

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