
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  $n$  be a positive integer. Alice writes  $n$  real numbers  $a_1, a_2,\dots, a_n$  in a line (in that order). Every move, she picks one number and replaces it with the average of itself and its neighbors ( $a_n$  is not a neighbor of  $a_1$ , nor vice versa). A number *changes sign* if it changes from being nonnegative to negative or vice versa. In terms of  $n$ , determine the maximum number of times that  $a_1$  can change sign, across all possible values of  $a_1,a_2,\dots, a_n$  and all possible sequences of moves Alice may make.

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