
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 three points $P(5,2)$, $F_{1}(-6,0)$, $F_{2}(6,0)$:  
(Ⅰ) Find the standard equation of the ellipse with foci $F_{1}$, $F_{2}$ that passes through point $P$;  
(Ⅱ) Let the symmetric points of $P$, $F_{1}$, $F_{2}$ with respect to the line $y=x$ be $P'$, $F_{1}'$, $F_{2}'$ respectively. Find the standard equation of the hyperbola with foci $F_{1}'$, $F_{2}'$ that passes through point $P'$.

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