
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
On a certain function calculator, there is a display and two operation buttons. If the first button is pressed, it changes the number on the display to $\left[\frac{x}{2}\right]$ (where $[x]$ denotes the largest integer not greater than the real number $x$). If the second button is pressed, it changes the number on the display to $4x + 1$. Pressing either button once is called an operation. Currently, the number on the display is 1. 

1. Is it possible to display the integer 2000 after a finite number of operations? Provide a reason.
2. How many integers less than 2000 can appear on the display after a finite number of operations?

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