
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
On the table, there are  $1024$  marbles and two students,  $A$  and  $B$ , alternatively take a positive number of marble(s). The student  $A$  goes first,  $B$  goes after that and so on. On the first move,  $A$  takes  $k$  marbles with  $1 < k < 1024$ . On the moves after that,  $A$  and  $B$  are not allowed to take more than  $k$  marbles or  $0$  marbles. The student that takes the last marble(s) from the table wins. Find all values of  $k$  the student  $A$  should choose to make sure that there is a strategy for him to win the game.

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