
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
Q. For integers  $m,n\geq 1$ , Let  $A_{m,n}$  ,  $B_{m,n}$  and  $C_{m,n}$  denote the following sets: $A_{m,n}=\{(\alpha _1,\alpha _2,\ldots,\alpha _m) \colon 1\leq \alpha _1\leq \alpha_2 \leq \ldots \leq \alpha_m\leq n\}$  given that  $\alpha _i \in \mathbb{Z}$  for all  $i$  $B_{m,n}=\{(\alpha _1,\alpha _2,\ldots ,\alpha _m) \colon \alpha _1+\alpha _2+\ldots + \alpha _m=n\}$  given that  $\alpha _i \geq 0$  and  $\alpha_ i\in \mathbb{Z}$  for all  $i$  $C_{m,n}=\{(\alpha _1,\alpha _2,\ldots,\alpha _m)\colon 1\leq \alpha _1< \alpha_2 < \ldots< \alpha_m\leq n\}$  given that  $\alpha _i \in \mathbb{Z}$  for all  $i$  $(a)$  Define a one-one onto map from  $A_{m,n}$  onto  $B_{m+1,n-1}$ . $(b)$  Define a one-one onto map from  $A_{m,n}$  onto  $C_{m,n+m-1}$ . $(c)$  Find the number of elements of the sets  $A_{m,n}$  and  $B_{m,n}$ .

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