
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 that the line $l$: $y=2x-1$ intersects with the hyperbola $\frac {x^{2}}{a^{2}}- \frac {y^{2}}{b^{2}}=1(a＞0， b＞0)$ at two distinct points $A$ and $B$, and $\overrightarrow {OA}\cdot \overrightarrow {OB}=0$ ($O$ is the origin).
(Ⅰ) Determine whether $\frac {1}{a^{2}}- \frac {1}{b^{2}}$ is a constant value, and explain the reason;
(Ⅱ) When the eccentricity $e$ of the hyperbola is in the range $( \sqrt {2}， \sqrt {3} )$, find the range of values for the length of the real axis of the hyperbola.

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