
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
For any positive integer  $n$ , define a function  $f$  by \[f(n)=2n+1-2^{\lfloor\log_2n\rfloor+1}.\] Let  $f^m$  denote the function  $f$  applied  $m$  times.. Determine the number of integers  $n$  between  $1$  and  $65535$  inclusive such that  $f^n(n)=f^{2015}(2015).$ *Proposed by Yannick Yao*


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