
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
In a geometric sequence with common ratio $q$, it is known that $a_1=16$, and $a_1$, $a_2+2$, $a_3$ form an arithmetic sequence.

$(1)$ Find $q$ and $a_n$;

$(2)$ If $q < 1$, find the minimum positive integer value of $n$ that satisfies $a_1-a_2+a_3-a_4+\cdots +(-1)^{2n-1}\cdot a_{2n} > 10$.

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