
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 $f(x)=|2x-9|-|x-5|$.<br/>$(1)$ Find the solution set of the inequality $f(x)\geqslant 2x-1$;<br/>$(2)$ The minimum value of the function $y=f(x)+3|x-5|$ is $m$. For positive real numbers $a$ and $b$ satisfying $\frac{1}{a}+\frac{3}{b}=m$, find the minimum value of $a+3b$.

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