
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
Let  $f:\mathbb{N}\to\mathbb{N}$  be a function satisfying the following conditions:

(a)  $f(1)=1$ .
(b)  $f(a)\leq f(b)$  whenever  $a$  and  $b$  are positive integers with  $a\leq b$ .
(c)  $f(2a)=f(a)+1$  for all positive integers  $a$ .

How many possible values can the  $2014$ -tuple  $(f(1),f(2),\ldots,f(2014))$  take?

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