
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
Initially, Alice is given a positive integer  $a_0$ . At time  $i$ , Alice has two choices,  $$ \begin{cases}a_i\mapsto\frac1{a_{i-1}}a_i\mapsto2a_{i-1}+1\end{cases} $$  Note that it is dangerous to perform the first operation, so Alice cannot choose this operation in two consecutive turns. However, if  $x>8763$ , then Alice could only perform the first operation. Determine all  $a_0$  so that  $\{i\in\mathbb N\mid a_i\in\mathbb N\}$  is an infinite set.

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