
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 an infinite sequence $\{a_n\}$ satisfying $a_{n+1} = p \cdot a_n + \frac{q}{a_n}$ ($n \in \mathbb{N}^*$), where $p$ and $q$ are non-negative real numbers and not both zero.  
(1) If $p = \frac{1}{2}$, $q = 2$, and $a_3 = \frac{41}{20}$, find the value of $a_1$;  
(2) If $a_1 = 5$, $p \cdot q = 0$, find the sum of the first $n$ terms of the sequence $\{a_n\}$, denoted as $S_n$;  
(3) If $a_1 = 2$, $q = 1$, and $\{a_n\}$ is a monotonically decreasing sequence, find the range of the real number $p$.

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