
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: $\sqrt{1+2\sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{1+\ldots}}}}}=3$, find the value of $\sqrt{1+3\sqrt{1+4\sqrt{1+5\sqrt{1+6\sqrt{1+\ldots}}}}}$; thereafter, calculate the values of $\sqrt{1+4\sqrt{1+5\sqrt{1+6\sqrt{1+7\sqrt{1+\ldots}}}}}$ and $\sqrt{1+5\sqrt{1+6\sqrt{1+7\sqrt{1+8\sqrt{1+\ldots}}}}}$, and make a conjecture for the general term $\sqrt{1+n\sqrt{1+(n+1)\sqrt{1+(n+2)\sqrt{1+(n+3)\sqrt{1+\ldots}}}}}$ where $n \in \mathbb{N}^*$.

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