
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 $f(x) = |x+2|$.
1. Solve the inequality $2f(x) < 4 - |x-1|$;
2. Given $m+n=1$ ($m>0$, $n>0$), if the inequality $|x-a|-f(x) \leq \frac{1}{m} + \frac{1}{n}$ always holds, find the range of the real number $a$.

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