
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
We have a calculator with two buttons that displays an integer \( x \). Pressing the first button replaces \( x \) by \( \left\lfloor \frac{x}{2} \right\rfloor \), and pressing the second button replaces \( x \) by \( 4x + 1 \). Initially, the calculator displays 0. How many integers less than or equal to 2014 can be achieved through a sequence of arbitrary button presses? (It is permitted for the number displayed to exceed 2014 during the sequence. Here, \( \lfloor y \rfloor \) denotes the greatest integer less than or equal to the real number \( y \).)

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