
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: $x^2+y^2=25$, and 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 $l$ and 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, explain why.

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