
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
There is a function calculator with a display screen and two operation keys. If the first operation key is pressed, the number $x$ on the display screen is changed to $\left[\frac{x}{2}\right]$ (where $[x]$ denotes the greatest integer less than or equal to $x$). If the second operation key is pressed, the number $x$ on the display screen is changed to $4x + 1$. Pressing either operation key is considered one operation. Currently, the number on the display screen is 1. 

(1) Can the number 2000 appear on the display screen after a finite number of operations? Provide reasoning.

(2) How many integers less than 2000 can appear on the display screen after a finite number of operations?



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