
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
Consider the function $f(x) = |x+4|.$  
(1) If $y = f(2x+a) + f(2x-a)$ has a minimum value of 4, find the value of $a$;  
(2) Find the solution set of the inequality $f(x) > 1-\frac{1}{2}x.$

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