
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
The set  $\mathcal{A}$  consists with some functions from  $\{1,2,\ldots,2023\}$  to  $\{1,2,\ldots,2023\}$  satisfying the following two conditions: 
- For any function  $f$  in  $\mathcal{A}$  and two integers  $x,y$  ( $1\leq x<y\leq 2023$ ), we have  $f(x)\geq f(y)$ .
- For any functions  $f,g$  in  $\mathcal{A}$  and any integer  $x$  ( $1\leq x\leq 2023$ ), we have  $f(g(x))=g(f(g(x)))$ . 
Find the maximal possible number of elements in  $\mathcal{A}$ . 

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