
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
Let $f(x)=ax^{2}+(2-a)x+a$.  
$(1)$ If the inequality $f(x)\geqslant 1$ holds for all real numbers $x$, find the range of real number $a$;  
$(2)$ Solve the inequality $f(x) \lt a+2$ where $a$ is a real number.

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