
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  $X$  be a  $5\times 5$  matrix with each entry be  $0$  or  $1$ . Let  $x_{i,j}$  be the  $(i,j)$ -th entry of  $X$  ( $i,j=1,2,\hdots,5$ ). Consider all the  $24$  ordered sequence in the rows, columns and diagonals of  $X$  in the following:
\begin{align*}
&(x_{i,1}, x_{i,2},\hdots,x_{i,5}),\ (x_{i,5},x_{i,4},\hdots,x_{i,1}),\ (i=1,2,\hdots,5) 
&(x_{1,j}, x_{2,j},\hdots,x_{5,j}),\ (x_{5,j},x_{4,j},\hdots,x_{1,j}),\ (j=1,2,\hdots,5) 
&(x_{1,1},x_{2,2},\hdots,x_{5,5}),\ (x_{5,5},x_{4,4},\hdots,x_{1,1}) 
&(x_{1,5},x_{2,4},\hdots,x_{5,1}),\ (x_{5,1},x_{4,2},\hdots,x_{1,5})
\end{align*}
Suppose that all of the sequences are different. Find all the possible values of the sum of all entries in  $X$ .

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