
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
Find all positive integers  $n \ge  2$  such that equality   $i+j \equiv C_{n}^{i} +  C_{n}^{j}$  (mod  $2$ ) is true for arbitrary  $0 \le  i  \le j  \le n$ .

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