
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

$$
\begin{aligned}
& x_{1}+2 x_{2}+2 x_{3}+\ldots+2 x_{100}=1 \\
& x_{1}+3 x_{2}+4 x_{3}+\ldots+4 x_{100}=2 \\
& x_{1}+3 x_{2}+5 x_{3}+\ldots+6 x_{100}=3 \\
& \ldots \\
& x_{1}+3 x_{2}+5 x_{3}+\ldots+199 x_{100}=100 .
\end{aligned}
$$

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