
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
The number  $1$  is written on the blackboard. After that a sequence of numbers is created as follows: at each step each number  $a$  on the blackboard is replaced by the numbers  $a - 1$  and  $a + 1$ ; if the number  $0$  occurs, it is erased immediately; if a number occurs more than once, all its occurrences are left on the blackboard. Thus the blackboard will show  $1$  after  $0$  steps;  $2$  after  $1$  step;  $1, 3$  after  $2$  steps;  $2, 2, 4$  after  $3$  steps, and so on. How many numbers will there be on the blackboard after  $n$  steps?

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