
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
Problem: Can you compare the sizes of $2023^{2024}$ and $2024^{2023}$?<br/>To solve this problem, we can start by simplifying the situation, trying to discover patterns from simple cases, then through induction, conjecture a conclusion.<br/>$(1)$ By calculation, compare the following sets of numbers:<br/>①$1^{2}$______$2^{1}$; ②$2^{3}$______$3^{2}$; ③$3^{4}$______$4^{3}$; ④$4^{5}$______$5^{4}$; ⑤$5^{6}$______$6^{5}\ldots $<br/>$(2)$ Based on the above results, conjecture the sizes of $2023^{2024}$ and $2024^{2023}$. (No need to explain the reason)<br/>$(3)$ If $a$ is a positive integer, based on the above conclusion, compare $a^{a+1}$ and $\left(a+1\right)^{a}$.

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