
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
Evin’s calculator is broken and can only perform  $3$  operations:
Operation  $1$ : Given a number  $x$ , output  $2x$ . 
Operation  $2$ : Given a number  $x$ , output  $4x +1$ . 
Operation  $3$ : Given a number  $x$ , output  $8x +3$ . 
After initially given the number  $0$ , how many numbers at most  $128$  can he make?


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