
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 \(C_{1}\) and a hyperbola \(C_{2}\) that share the same foci \(F_{1}(3,0)\) and \(F_{2}(-3,0)\), and for which the minor axis of the ellipse coincides with the conjugate axis of the hyperbola. Determine the number of lattice points inside the region enclosed by the intersection points of \(C_{1}\) and \(C_{2}\).

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