
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
If the solution of a linear inequality (group) ① is also the solution of a linear inequality (group) ②, then the linear inequality ② is called the "covering inequality" of the linear inequality ①. In particular, if an inequality (group) has no solution, then it is covered by any other inequality (group). For example, the solution of the inequality $x \gt 1$ is also the solution of the inequality $x\geqslant -1$, then $x\geqslant -1$ is the "covering inequality" of $x \gt 1$. The inequality system $\left\{\begin{array}{l}{x＜3}\\{x≥5}\end{array}\right.$ has no solution, then any other inequality (group) is its "covering inequality". Based on the above information, solve the following problems:  
$(1)$ The "covering inequality" of $x \lt -1$ with $x \lt -3$ is (fill in "is" or "is not");  
$(2)$ If $x \lt -2$ is the "covering inequality" of the inequality $-x+4m \gt 0$ with respect to $x$, find the range of values for $m$;  
$(3)$ If the inequality system $\left\{\begin{array}{l}{2a-x＞1}\\{2x+5＞3a}\end{array}\right.$ is covered by $1\leqslant x\leqslant 6$, find the range of values for $a$.

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