
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
Consider a sequence of positive terms $\{a_n\}$, satisfying $a_1=1$, $a_2= \dfrac{1}{2}$, and $\dfrac{1}{a_{n+1}}= \sqrt{\dfrac{1}{a_n}\cdot \dfrac{1}{a_{n+2}}} \ (n\in\mathbb{N}^*)$. Determine the value of $a_1 \cdot a_3 + a_2 \cdot a_4 + a_3 \cdot a_5 + \ldots + a_n \cdot a_{n+2}$.

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