
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
Let $M \subseteq \mathbb{N}^{+}$, and for the positive sequence $\{a_n\}$, the product of its first $n$ terms is $T_n$. It is given that for all $k \in M$, when $n > k$, the equation $\sqrt{T_{n+k}T_{n-k}} = T_nT_k$ holds.  
$(1)$ If $M = \{1\}$, $a_1 = \sqrt{3}$, $a_2 = 3\sqrt{3}$, find the sum of the first $n$ terms of the sequence $\{a_n\}$.  
$(2)$ If $M = \{3,4\}$, $a_1 = \sqrt{2}$, find the general formula for the sequence $\{a_n\}$.

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