
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 ellipse $C$: $\frac{x^2}{a^2}+\frac{y^2}{b^2}=1 (a>b>0)$ and ellipse $\frac{x^2}{5}+\frac{y^2}{2}=1$ have the same foci. A chord passing through the right focus of ellipse $C$ and perpendicular to the x-axis has a length of 1. <br/> $(1)$ Find the equation of ellipse $C$; <br/> $(2)$ Line $l: y=x+m$ intersects ellipse $C$ at points $A$ and $B$. If $|AB|=\frac{8}{5}$, find the value of the real number $m$.

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