
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 a quadratic function $f(x)$ with the axis of symmetry $x=-2$, the chord length intercepted by the $x$-axis of the graph of $f(x)$ is $2 \sqrt{3}$, and it satisfies $f(0)=1$.

$(1)$ Find the analytical expression of $f(x)$;

$(2)$ If $f\left( \left( \dfrac{1}{2} \right)^{x} \right) > k$ holds true for $x \in [-1,1]$, find the range of the real number $k$.

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