
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  $n$ , consider a triangular array with entries  $a_{ij}$  where  $i$  ranges from  $1$  to  $n$  and  $j$  ranges from  $1$  to  $n-i+1$ . The entries of the array are all either  $0$  or  $1$ , and, for all  $i > 1$  and any associated  $j$  ,  $a_{ij}$  is  $0$  if  $a_{i-1,j} = a_{i-1,j+1}$ , and  $a_{ij}$  is  $1$  otherwise. Let  $S$  denote the set of binary sequences of length  $n$ , and define a map  $f \colon S \to S$  via  $f \colon (a_{11}, a_{12},\cdots ,a_{1n}) \to (a_{n1}, a_{n-1,2}, \cdots , a_{1n})$ . Determine the number of fixed points of  $f$ .

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