
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 the parabola $E: y^2 = 2px (p > 0)$ with focus $F$, and a fixed point $M(2,3)$ on the opposite side of $F$ with respect to the parabola $E$. The minimum sum of the distances from a moving point $P$ on the parabola $E$ to the point $M$ and to its directrix $l$ is $\sqrt{10}$.

$(1)$ Find the equation of the parabola $E$;

$(2)$ Suppose the line $y= \frac{1}{2}x+b$ intersects the circle $x^2+y^2=9$ and the parabola $E$ at four distinct points, from left to right, denoted as $A, B, C, D$, with $B, D$ being the intersection points with the parabola $E$. If the slopes of lines $BF$ and $DF$ are complementary angles, find the value of $|AB|+|CD|$.

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