
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
Let $f(x) = |x-1| + 2|x+1|$ be the minimum value of $m$.  
(Ⅰ) Find the value of $m$;  
(Ⅱ) Let $a, b \in \mathbb{R}$, and $a^2 + b^2 = m$, find the minimum value of $\frac{1}{a^2 + 1} + \frac{4}{b^2 + 1}$.

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