
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
Given $f(1,1)=1$, $f(m,n) \in \mathbb{N}^*$ for $m,n \in \mathbb{N}^*$, and for any $m,n \in \mathbb{N}^*$:
1. $f(m, n+1) = f(m, n) + 3$;
2. $f(m+1, 1) = 2f(m, 1)$.

Considering the following propositions:
(i) Sequence $\{f(m,2015)\}$ is a geometric sequence;
(ii) Sequence $\{f(2015,n)\}$ is an arithmetic sequence;
(iii) $f(1,1) + f(1,2) + \ldots + f(1,2015) = 2^{2015} - 1$;
(iv) $f(1,1) + f(2,1) + \ldots + f(2015,1) = 2^{2015} - 1$.

The correct propositions are: _______.

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