
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 the set of integers
\[ A = \{1, 2, \cdots, 100\} \]

let the function \( f: A \rightarrow A \) satisfy:
1. For any \( 1 \leqslant i \leqslant 99 \), 
\[ |f(i) - f(i+1)| \leqslant 1 \]
2. For any \( 1 \leqslant i \leqslant 100 \), 
\[ f(f(i)) = 100 \]
Find the minimum possible value of \(\sum_{i=1}^{100} f(i)\).

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