
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 a line $l$ on a plane and a point $P$ at distance 1 from $l$, let $A_{1}, A_{2}, \cdots, A_{100}$ be 100 points on $l$ such that
$$
\begin{array}{l}
\overrightarrow{P A_{i}} \cdot \overrightarrow{P A_{i+1}}=-\frac{2}{i} \quad (i=1,2,\cdots,99). \\
\text{Then, } \overrightarrow{P A_{100}} \cdot \overrightarrow{P A_{1}}=
\end{array}
$$

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