
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 ${A_1}=\sqrt{({n+3})({n-1})+4}$, ${A_2}=\sqrt{({n+5}){A_1}+4}$, ${A_3}=\sqrt{({n+7}){A_2}+4}$, ${A_4}=\sqrt{({n+9}){A_3}+4}$, $\ldots {A_k}=\sqrt{({n+2k+1}){A_{k-1}}+4}$, $\ldots $, and $A_{100}=2005$, find the value of $n$.

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