
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 positive sequence $\{a_{n}\}$, $a_{1}=1$, $a_{n+1}^2-a_n^2=1$. Let ${b_n}=\frac{1}{{({{a_n}+1})({{a_{n+1}}+1})({{a_n}+{a_{n+1}}})}}$. Determine the sum of the first $m$ terms of the sequence $\{b_{n}\}$, where $\lg (10^{119}+1) \lt m \lt \lg (10^{120}+1)$.

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