
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 the function $f(x)=a(|\sin x|+|\cos x|)+4\sin 2x+9$, where $f\left(\frac{9\pi}{4}\right)=13-9\sqrt{2}$. Find the value of $a$ and determine the smallest positive period of $y=f(x)$ without proof. Find the number of zeros of $y=f(x)$ on the interval $[0,\frac{\pi}{2}]$. Determine whether there exists a positive integer $n$ such that $y=f(x)$ has exactly 2022 zeros on the interval $[0,\frac{n\pi}{4}]$. If it exists, find the value of $n$; if not, explain the reason.

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