
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 and integer  $x$ . Pressing the first button replaces  $x$  by  $\lfloor \frac{x}{2} \rfloor$ , 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}
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