
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 positive sequence $\{a_n\}$ where $a_1=t$, and the sum of its first $n$ terms is $S_n$, satisfying $2S_n=a_n \cdot a_{n+1}$.  
(1) If the sequence $\{a_n\}$ is an arithmetic sequence, find the value of $t$ and derive the general formula for the sequence $\{a_n\}$.  
(2) If the sequence $\{a_n\}$ is a monotonically increasing sequence, find the range of values for $t$.

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