
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 the function $f(x)=|2x-a|+|x+a|$ where $(a > 0)$.  
$(1)$ When $a=1$, find the minimum value of $f(x)$;  
$(2)$ If the inequality $f(x) < \frac{5}{x}+a$ has a solution for $x \in [1,2]$, find the range of the real number $a$.

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