
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 $f(x) = a^{x-1} (x \geqslant 0)$, where $a > 0$ and $a \neq 1$.

$(1)$ If the graph of $x \geqslant 0$ passes through the point $(2, \frac{1}{2})$, find the value of $a$;

$(2)$ Find the range of the function $y = f(x) (x \geqslant 0)$.

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