
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 system of equations:

$$
\left\{\begin{aligned}
& x_{1}+2 x_{2}+3 x_{3}+x_{4}=1 \\
& 3 x_{1}+13 x_{2}+13 x_{3}+5 x_{4}=3 \\
& 3 x_{1}+7 x_{2}+7 x_{3}+2 x_{4}=12 \\
& x_{1}+5 x_{2}+3 x_{3}+x_{4}=7 \\
& 4 x_{1}+5 x_{2}+6 x_{3}+x_{4}=19
\end{aligned}\right.
$$

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