
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 function $f(x)= \begin{cases} \log_2 x, & x\geqslant 1 \\ -\log_2 x, & 0 < x < 1 \end{cases}$, if $f(a)=f(b)$ ($0 < a < b$), find the minimum value of $\frac{1}{a}+\frac{4}{b}$ and the value of $f(a+b)$.

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