
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 circle $O: x^{2}+y^{2}=25$, the center of circle $O_{1}$ is $O_{1}(m,0)$, circle $O$ and circle $O_{1}$ intersect at point $P(3,4)$. A line $l$ passing through point $P$ with a slope of $k (k \neq 0)$ intersects circles $O$ and $O_{1}$ at points $A$ and $B$, respectively.  
$(1)$ If $k=1$ and $|BP|=7 \sqrt {2}$, find the equation of circle $O_{1}$;  
$(2)$ A line $l_{1}$ perpendicular to $l$ passing through point $P$ intersects circles $O$ and $O_{1}$ at points $C$ and $D$, respectively. When $m$ is a constant, determine whether $|AB|^{2}+|CD|^{2}$ is a constant value. If so, find this constant value; if not, please explain why.

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