chore: update @eslint/core, add legacy-peer-deps, and use ESLint v10

[lumirlumir]

Prerequisites checklist

• I have read the contributing guidelines.

What is the purpose of this pull request?

This PR was motivated by eslint/json#187 (comment).

Currently, when updating @eslint/core to v1.0.0, CI fails because @eslint/core@1.0.0 does not match ESLint v9

@eslint/core@1.0.0 includes type information for ESLint v10, which causes the type mismatch.

So, I've updated ESLint to >=10.0.0-alpha.0 <10.0.0 || ^10.0.1. (Ref: https://github.com/eslint/eslint/blob/main/packages/eslint-config-eslint/package.json#L66)

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When I update ESLint to >=10.0.0-alpha.0 <10.0.0 || ^10.0.1, the following error occurs, so I added the legacy-peer-deps option to .npmrc.

This change follows the approach in eslint/eslint#20281

What changes did you make? (Give an overview)

In this PR, I've updated @eslint/core, add legacy-peer-deps, and use ESLint v10 pre-release.

Related Issues

Ref: #327, eslint/json#187 (comment), eslint/eslint#20281

Is there anything you'd like reviewers to focus on?

N/A

[lumirlumir]

Counterpart PR in eslint/json: eslint/json#190

[mdjermanovic]

Same question as eslint/json#190 (comment). We can clarify there, just wanted to leave a note here as well so that we don't merge until this is clarified.