
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
In the sequence $\{a_{n}\}$, $a_{1}=1$, $\sqrt{{a}_{n+1}}-\sqrt{{a}_{n}}=1$ ($n\in N^{*}$); the sum of the first $n$ terms of a geometric sequence $\{b_{n}\}$ is $S_{n}=2^{n}-m$. For $n\in N^{*}$, the smallest value of the real number $\lambda$ that satisfies $\lambda b_{n}\geqslant a_{n}$ for all $n$ is ______.

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