
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 an arithmetic sequence \\(\{a_n\}\) and a geometric sequence \\(\{b_n\}\) where \\(a_1=b_1=1\\), \\(a_2=b_2\\), and \\(a_4+2=b_3\\).  
(I) Find the general formula for sequences \\(\{a_n\}\) and \\(\{b_n\}\);  
(II) If \\(a_m=b_n (n\in\mathbb{N}^*)\\), write the relationship between \\(m\\) and \\(n\\) as \\(m=f(n)\\), and calculate \\(f(1)+f(2)+\ldots+f(n)\\).

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