
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 focus of the parabola $C:y^{2}=4x$ is $F$, and the line $l$ passing through $F$ with a slope of $-1$ intersects the parabola $C$ at points $A$ and $B$, the equation of the circle $D$ with diameter $AB$ is ____; if there exist two points $P$ and $Q$ on circle $D$, and a point $M$ on circle $T:\left(x+2\right)^{2}+\left(y+7\right)^{2}=a^{2}(a \gt 0)$ such that $\angle PMQ=90^{\circ}$, then the range of the real number $a$ is ____.

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