
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
Let  $n$  be a natural number greater than 2.  $l$  is a line on a plane. There are  $n$  distinct points  $P_1$ ,  $P_2$ , …,  $P_n$  on  $l$ . Let the product of distances between  $P_i$  and the other  $n-1$  points be  $d_i$  ( $i = 1, 2,$  …,  $n$ ). There exists a point  $Q$ , which does not lie on  $l$ , on the plane. Let the distance from  $Q$  to  $P_i$  be  $C_i$  ( $i = 1, 2,$  …,  $n$ ). Find  $S_n = \sum_{i = 1}^{n} (-1)^{n-i} \frac{c_i^2}{d_i}$ .

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