
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
Given $n$ parallel lines $l_1: x - y + C_1 = 0, C_1 = \sqrt{2}$, $l_2: x - y + C_2 = 0$, $l_3: x - y + C_3 = 0$, ..., $l_n: x - y + C_n = 0$ (where $C_1 < C_2 < C_3 < ... < C_n$), the distance between each pair of consecutive lines from these $n$ lines increases successively by 2, 3, 4, ..., $n$.
1. Find $C_n$;
2. Find the area of the shape formed by $l_n : x - y + C_n = 0$ and the $x$-axis and $y$-axis;
3. Find the area of the shape formed by $l_{n-1} : x - y + C_{n-1} = 0$, $l_n : x - y + C_n = 0$, and the $x$-axis and $y$-axis.

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