
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 of positive terms $\{a_{n}\}$ satisfying $a_{1}=1$, $a_{2}=2$, $a_{4}=64$, and ${a_n}{a_{n+2}}=ka_{n+1}^2$ for $n\in\mathbb{N}^*$.  
$(1)$ Find the value of $k$;  
$(2)$ Determine the general formula for the sequence $\{a_{n}\}$.

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