
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 $f(x) = x^2 - (a+2)x + 4$ $(a \in \mathbb{R})$.  
$(1)$ Solve the inequality with respect to $x$: $f(x) \leq 4 - 2a$;  
$(2)$ If for any $x \in [1,4]$, $f(x) + a + 1 \geq 0$ always holds, find the range of real numbers for $a$;  
$(3)$ Given $g(x) = mx + 5 - 2m$, when $a = 2$, if for any $x_{1} \in [1,4]$, there always exists $x_{2} \in [1,4]$ such that $f(x_{1}) = g(x_{2})$, find the range of real numbers for $m$.

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