
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
Regarding the system of equations for \( x \), \( y \), \( z \), and \( w \):
$$
\left\{\begin{array}{l}
x^{2} + 2 y^{2} + 2 z^{2} + w^{2} = 43, \\
y^{2} + z^{2} + w^{2} = 29, \\
5 z^{2} - 3 w^{2} + 4 x y + 12 y z + 6 z x = 95
\end{array}\right.
$$
find the set of solutions.

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