
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 sequence $\{a_n\}$ that satisfies $P_n(n, a_n)$ with $\overrightarrow{P_nP_{n+1}} = (1, 2)$ for any $n \in \mathbb{N}^*$, and $a_1 + a_2 = 4$, find the sum $S_n$ of the first $n$ terms of the sequence $\{\frac{1}{a_n \cdot a_{n+1}}\}$.

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