
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 $y=f(x)$ is a quadratic function, and the equation $f(x)=0$ has two equal real roots, and $f'(x)=2x+2$.

$(1)$ Find the expression for $f(x)$.

$(2)$ Find the area of the shape formed by the function $y=f(x)$ and the function $y=-x^{2}-4x+1$.

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