
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 $f(x)=e^x- \frac{x^2}{2}-ax-1$, where $a$ is a real number.

$(1)$ If $a=-\frac{1}{2}$, find the equation of the tangent line to the curve $y=f(x)$ at the point $(1,f(1))$;

$(2)$ When $x\geqslant \frac{1}{2}$, if the inequality $f(x)\geqslant 0$ always holds for $x$, find the range of values for $a$.

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