
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 $\{a_n\}$ satisfying: $a_1= \frac {1}{2}$, $a_{n+1}-a_n=p\cdot3^{n-1}-nq$, where $n\in\mathbb{N}^*$, $p$, $q\in\mathbb{R}$.  
(1) If $q=0$ and the sequence $\{a_n\}$ is a geometric sequence, find the value of $p$;  
(2) If $p=1$ and $a_4$ is the minimum term of the sequence $\{a_n\}$, find the range of $q$.

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