Typing shadowRoot of LitElement

[Jaewoook]

Is your feature request related to a problem? Please describe. A clear and concise description of what the problem is. Ex. I'm always frustrated when [...] 기능 요청이 특정 문제에 연관된 것이라면 여기에 그 문제를 설명해주세요. Currently, LitElement's shadowRoot property is typed as any in entire project. So it makes all variables from shadowRoot's result type to any. As a result, possible type error could occur anytime and code are not type-safe. (TypeScript backend could not type-check properly)

I think we need to remove shadowRoot: any and make type-safe code.

This issue is sub-task of #1354

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type errors when removing shadowRoot: any in src/components/backend-ai-pages.ts