
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

$$
\left\{\begin{array}{rlrl}
10 x_{1}+3 x_{2}+4 x_{3}+x_{4}+x_{5} & & = 0 \\
11 x_{2}+2 x_{3}+2 x_{4}+3 x_{5}+x_{6} & = 0 \\
15 x_{3}+4 x_{4}+5 x_{5}+4 x_{6}+x_{7} & = 0 \\
2 x_{1}+x_{2} - 3 x_{3} + 12 x_{4} - 3 x_{5} + x_{6} + x_{7} & = 0 \\
6 x_{1} - 5 x_{2} + 3 x_{3} - x_{4} + 17 x_{5} + x_{6} & = 0 \\
3 x_{1} + 2 x_{2} - 3 x_{3} + 4 x_{4} + x_{5} - 16 x_{6} + 2 x_{7} & = 0 \\
4 x_{1} - 8 x_{2} + x_{3} + x_{4} - 3 x_{5} & + 19 x_{7} & = 0
\end{array}\right.
$$

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