
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
Let point $F$ be the left focus of the ellipse $ C: \frac {x^{2}}{4m}+ \frac {y^{2}}{3m}=1 \ (m>0)$, and the line $y=x$ intersects the ellipse $C$ to form a chord with length $\frac {4 \sqrt {42}}{7}$.
(1) Find the equation of ellipse $C$;
(2) Circle $P: (x+ \frac {4 \sqrt {3}}{7})^{2}+(y- \frac {3 \sqrt {3}}{7})^{2}=r^{2} \ (r>0)$ intersects ellipse $C$ at points $A$ and $B$, with $M$ being any point on segment $AB$. The line through $F$ and $M$ intersects the ellipse $C$ at points $P$ and $Q$. $AB$ is the diameter of circle $P$, and the slope of line $FM$ is greater than 1. Find the range of values for $|PF|\cdot|QF|$.

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