
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 $f(x)=|x-4|-|x+2|$.<br/>$(1)$ If $f(x)-a^{2}+5a\geqslant 0$ always holds, find the range of values for $a$;<br/>$(2)$ If the maximum value of $f(x)$ is $M$, and positive real numbers $a$, $b$, $c$ satisfy $a+b+c=M$, find the maximum value of $\sqrt{a+1}+\sqrt{b+2}+\sqrt{c+3}$.

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