
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  $x_1,\ldots,x_n$  be a sequence with each term equal to  $0$  or  $1$ . Form a triangle as follows: its first row is  $x_1,\ldots,x_n$  and if a row if  $a_1, a_2, \ldots, a_m$ , then the next row is  $a_1 + a_2, a_2 + a_3, \ldots, a_{m-1} + a_m$  where the addition is performed modulo  $2$  (so  $1+1=0$ ). For example, starting from  $1$ ,  $0$ ,  $1$ ,  $0$ , the second row is  $1$ ,  $1$ ,  $1$ , the third one is  $0$ ,  $0$  and the fourth one is  $0$ .

A sequence is called good it is the same as the sequence formed by taking the last element of each row, starting from the last row (so in the above example, the sequence is  $1010$  and the corresponding sequence from last terms is  $0010$  and they are not equal in this case). How many possibilities are there for the sequence formed by taking the first element of each row, starting from the last row, which arise from a good sequence?  

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