
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  $3 \times 3$  symmetric real matrix  $A$ , we define  $f(A)$  as a  $3 \times 3$  matrix with the same eigenvectors of  $A$  such that if  $A$  has eigenvalues  $a$ ,  $b$ ,  $c$ , then  $f(A)$  has eigenvalues  $b+c$ ,  $c+a$ ,  $a+b$  (in that order). We define a sequence of symmetric real  $3\times3$  matrices  $A_0, A_1, A_2, \ldots$  such that  $A_{n+1} = f(A_n)$  for  $n \geq 0$ . If the matrix  $A_0$  has no zero entries, determine the maximum number of indices  $j \geq 0$  for which the matrix  $A_j$  has any null entries.

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