
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 $\{a_{n}\}$ where ${a_1}=1,{a_2}=2$, and $\frac{n}{2}{a_{n+2}}$ is an arithmetic mean of $({n+2}){a_n},λ({{n^2}+2n})$, if ${a_{2n+1}}>{a_{2n}}$ ($n∈{N^*}$), then the range of real number $\lambda$ is ______.

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