
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
For integers $a$ and $b$, define the operation as follows: $a$※$b=(a^{b})^{m}+(b^{a})^{n}$ (where $m$ and $n$ are constants), such as $3$※$2=(3^{2})^{m}+(2^{3})^{n}$.  
$(1)$ Fill in the blank: when $m=1$, $n=2023$, $2$※$1=$______.  
$(2)$ If $1$※$4=10$, $2$※$2=15$, find the value of $4^{2m+n-1}$.

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