
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
Two people,  $A$  and  $B$ , play the following game:  $A$  start choosing a positive integrer number and then, each player in it's turn, say a number due to the following rule:

If the last number said was odd, the player add  $7$  to this number;
If the last number said was even, the player divide it by  $2$ .

The winner is the player that repeats the first number said. Find all numbers that  $A$  can choose in order to win. Justify your answer.

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