
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 $f(x)=|x+a|+|2x-a|$.
- $(1)$ When $a=1$, solve the inequality $f(x) < 2$ for its solution set;
- $(2)$ Given $m+n=1$, for any $m$, $n\in(0,+\infty)$, there always exists $a\in\mathbb{R}$, such that $f(x)\leqslant \frac {1}{m}+ \frac {4}{n}$, find the maximum value of $x$.

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