
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 an ellipse centered at the origin with its foci on the x-axis and an eccentricity of $\dfrac{\sqrt{5}}{5}$. If the focus of the parabola $y^{2}=4x$ coincides with one focus of the ellipse.  
$(1)$ Find the standard equation of the ellipse.  
$(2)$ If a line $m$ passes through the left focus $F_{1}$ of the ellipse with a slope of $1$, and intersects the ellipse at points $A$ and $B$, find the length of the chord $|AB|$.

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