
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
Can you compare the magnitude of $2004^{2005}$ and $2005^{2004}$? To solve this problem, first write it in the general form, i.e., compare the magnitude of $n^{n+1}$ and $(n+1)^n$ (where $n$ is a positive integer), then start by analyzing the simple cases such as $n=1$, $n=2$, $n=3$, ..., and try to discover the pattern. After induction, make conjectures about the conclusion.  
By calculation, compare the magnitude of the following pairs of numbers (fill in ">" or "<" on the line):  
$1^2\quad 2^1$, $2^3\quad 3^2$, $3^4\quad 4^3$, $4^5\quad 5^4$, $5^6\quad 6^5$, ...

(2) From the results of question (1), what can you guess about the relation between $n^{n+1}$ and $(n+1)^n$?  
(3) Based on the inductive conjecture made above, try to compare the magnitude of $2004^{2005}$ and $2005^{2004}$.

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