
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 $a_{1}, a_{2}, \cdots, a_{10}$ and $b_{1}, b_{2}, \cdots, b_{10}$ are permutations of $1, 2, \cdots, 10 $, specifically
$$
\begin{array}{l}
\left\{a_{1}, a_{2}, \cdots, a_{10}\right\} \\
=\left\{b_{1}, b_{2}, \cdots, b_{10}\right\} \\
=\{1, 2, \cdots, 10\}.
\end{array}
$$

Let $c_{k}=a_{k}^{2} + b_{k}^{2} (k=1,2, \cdots, 10)$. Suppose the maximum value among $c_{1}, c_{2}, \cdots, c_{10}$ is $M$, and the minimum value is $m$. Find:
(1) The minimum possible value of $M$;
(2) The maximum possible value of $m$.

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