
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 \( f: \mathbb{N} \rightarrow \mathbb{N} \) 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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