
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  $n \geq 5$  be a positive integer. There are  $n$  stars with values  $1$  to  $n$ , respectively. Anya and Becky play a game. Before the game starts, Anya places the  $n$  stars in a row in whatever order she wishes. Then, starting from Becky, each player takes the left-most or right-most star in the row. After all the stars have been taken, the player with the highest total value of stars wins; if their total values are the same, then the game ends in a draw. Find all  $n$  such that Becky has a winning strategy.
*Proposed by Ho-Chien Chen*

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