
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 $\{a_{n}\}$ that satisfies ${a}_{n+1}={a}_{n}^{2}+k,n∈{N}_{+},k∈R$.<br/>$(1)$ If $a_{1}=1$ and the sequence $\{a_{n}\}$ is an increasing sequence, find the range of real number $k$;<br/>$(2)$ If $a_{1}=3$ and $k=0$, find the general formula of the sequence $\{a_{n}\}$.

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