
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 $f(x) = e^{x-a} - x$ where $a \in \mathbb{R}$ and $e$ is the base of the natural logarithm.  
$(1)$ If the function $f(x)$ has two zeros, find the range of real numbers for $a$.  
$(2)$ Let $g(x) = e^{x-a} - x\ln x + (1-a)x$, where $a \in (1, 3 - \ln 3]$. Denote the local minimum of $g(x)$ as $h(a)$. Find the range of values for the function $h(a)$.

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