
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 the identity: $\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{1+\cdots}}}}}=3$, we can find the value of $\sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{1+6\sqrt{1+\cdots}}}}}$, and further calculate the values of $\sqrt{1+4\sqrt{1+5\sqrt{1+6\sqrt{1+7\sqrt{1+\cdots}}}}}$, $\sqrt{1+5\sqrt{1+6\sqrt{1+7\sqrt{1+8\sqrt{1+\cdots}}}}}$, and so on. From this, we can inductively conjecture that $\sqrt{1+n\sqrt{1+\left( n+1 \right)\sqrt{1+\left( n+2 \right)\sqrt{1+\left( n+3 \right)\sqrt{1+\cdots}}}}}=$                    $(n\in {{N}^{*}})$

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