
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 a hyperbola $E: \frac {x^{2}}{a^{2}} - \frac {y^{2}}{b^{2}} = 1$ with a focal distance of 4, a circle with the radius equal to the real semi-axis and centered at the origin is tangent to the line $x - y + \sqrt{6} = 0$.
 (Ⅰ) Find the equation of the hyperbola E;
 (Ⅱ) Let point F be the left focus of the hyperbola E. Is there a fixed point M on the x-axis from which any line l intersects the hyperbola E at points P and Q, such that $\overrightarrow {FP} \cdot \overrightarrow{FQ}$ is constant? If there is, determine this constant value and coordinates of all such fixed points M; if not, explain why.

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