
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 $f(x)=\frac{1}{3}|x-a|$, where $a\in \mathbb{R}$.  
$(1)$ When $a=2$, find the solution to the inequality $|x-\frac{1}{3}|+f(x)\geqslant 1$.  
$(2)$ Let the solution set of the inequality $|x-\frac{1}{3}|+f(x)\leqslant x$ be $M$. If $[\frac{1}{3}, \frac{1}{2}]\subseteq M$, find the range of real numbers for $a$.

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