
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 that \(a_{1}, a_{2}, a_{3}, a_{4}\) are positive real numbers, satisfying:
$$
\sum_{i=1}^{4} i a_{i} \leqslant 10, \quad a_{i} \geqslant \frac{1}{2^{4-i}} \quad (i=1, 2, 3, 4).
$$
Define \(f=\sum_{i=1}^{4} \frac{1}{1+a_{i}^{i}}\). Find the minimum value of \(f\).

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