
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
For  $n \in \mathbb{N}$ , consider  non-negative  valued functions  $f$   on  $\{1,2, \cdots , n\}$  satisfying  $f(i) \geqslant f(j)$   for  $i>j$   and  $\sum_{i=1}^{n} (i+ f(i))=2023.$  Choose  $n$  such that  $\sum_{i=1}^{n} f(i)$  is at least.  How many such functions  exist in that case?

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