
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 $f(x)=|x+a|+|2x+1|$, where $a\in\mathbb{R}$.
$(1)$ When $a=1$, find the solution set of the inequality $f(x)\leqslant 1$;
$(2)$ Let the solution set of the inequality $f(x)\leqslant -2x+1$ with respect to $x$ be $P$, and $\left[-1,-\frac{1}{4}\right]\subseteq P$, find the range of values for $a$.

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