
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 following system of equations:

$$
\begin{gathered}
a^{3} x + a y + z = a^{2} \\
x + y + z = 1 \\
8 x + 2 y + z = 4 
\end{gathered}
$$

What cases arise when \( a = 1 \), \( a = 2 \), and \( a = -3 \)?

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