
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
Given a positive integer  $n$ . A triangular array  $(a_{i,j})$  of zeros and ones, where  $i$  and  $j$  run through the positive integers such that  $i+j\leqslant n+1$  is called a *binary anti-Pascal  $n$ -triangle* if  $a_{i,j}+a_{i,j+1}+a_{i+1,j}\equiv 1\pmod{2}$  for all possible values  $i$  and  $j$  may take on. Determine the minimum number of ones a binary anti-Pascal  $n$ -triangle may contain.

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