
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 $C$ with foci at $F_{1}(-2,0)$ and $F_{2}(2,0)$, and the ellipse $C$ passes through the point $A\left(0, \sqrt{5}\right)$.

$(1)$ Find the standard equation of the ellipse $C$.

$(2)$ If a line $l$ passing through the left focus $F_{1}$ with a slope of $45^{\circ}$ intersects the ellipse $C$ at points $P$ and $Q$, find the length of the segment $PQ$.

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