
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 $f(x)=|x-a|+|x+2|$.
$(1)$ When $a=1$, solve the inequality $f(x)\leqslant 5$ for its solution set;
$(2)$ There exists $x_{0}\in \mathbb{R}$, such that $f(x_{0})\leqslant |2a+1|$, find the range of values for $a$.

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