
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
Let  $T$  be a positive integer. Find all functions  $f: \mathbb {Z}^+ \times \mathbb {Z}^+ \to \mathbb {Z}^+$ , such that there exists integers  $C_0,C_1,\ldots ,C_T$  satisfying:
(1) For any positive integer  $n$ , the number of positive integer pairs  $(k,l)$  such that  $f(k,l)=n$  is exactly  $n$ .
(2) For any  $t=0,1,\ldots ,T,$  as well as for any positive integer pair  $(k,l)$ , the equality  $f(k+t,l+T-t)-f(k,l)=C_t$  holds.

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