
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 equations:
\[ x_1 + 2x_2 + 3x_3 + \ldots + (n-1)x_{n-1} + nx_n = n \]
\[ 2x_1 + 3x_2 + 4x_3 + \ldots + nx_{n-1} + x_n = n-1 \]
\[ 3x_1 + 4x_2 + 5x_3 + \ldots + x_{n-1} + 2x_n = n-2 \]
\[ \ldots \]
\[ (n-1)x_1 + nx_2 + x_3 + \ldots + (n-3)x_{n-1} + (n-2)x_n = 2 \]
\[ nx_1 + x_2 + 2x_3 + \ldots + (n-2)x_{n-1} + (n-1)x_n = 1 \]

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