
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 $f(x) = 1 - \frac{4}{2a^x + a}$ $(a > 0, a \neq 1)$ and $f(0) = 0$.

$(1)$ Find the value of $a$;

$(2)$ If the function $g(x) = (2^x + 1)f(x) + k$ has a root, find the range of the real number $k$.

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