
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 that the general term of the sequence $\{a_{n}\}$ is ${a_n}=\frac{1}{{n({n+1})}}$, and the sum of its first $n$ terms is $S_{n}$. Let ${b_n}=\frac{λ}{{1-{S_n}}}+{n^2}$. If the sequence $\{b_{n}\}$ is strictly increasing, then the range of real number $\lambda$ is ______.

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