Determine and print expression to reach target

You are given four numbers and a target T (e.g., 24). Using +, −, ×, ÷ and parentheses, and using each number exactly once, determine whether you can form an expression that evaluates to T. Return a boolean. Follow-up: if it is possible, output one valid expression string. Clarify how to handle division by zero and floating-point tolerance; discuss algorithmic approach (e.g., backtracking with pruning), complexity, and how you would extend the method to N numbers or different operator sets.

Reach Target with Four Operations

You are given a list of numbers nums and a target value target. Using the four arithmetic operators +, -, *, / and parentheses, and using each number in nums exactly once, determine whether you can build an expression that evaluates to target.

Return true if such an expression exists, otherwise false. The classic instance of this problem is the "24 game" (four numbers, target 24).

Rules and clarifications:

• Each number must be used exactly once; numbers may be combined in any order.
• Any intermediate result may be a non-integer (division can produce fractions), and intermediate/final results are compared with a floating-point tolerance of 1e-6.
• Division by zero is not allowed: you may only divide by a value whose absolute value exceeds the tolerance.
• An empty input cannot reach any target, so return false.
• A single number x reaches target only if x equals target (within tolerance).

This generalizes naturally to N numbers and to arbitrary operator sets; the reference approach is backtracking that repeatedly collapses two values into one until a single value remains.

Constraints

• Each number in nums must be used exactly once.
• Allowed operators: + - * / and parentheses (any grouping).
• Intermediate and final values are compared with a floating-point tolerance of 1e-6.
• Division is only permitted when the divisor's absolute value exceeds 1e-6 (no division by zero).
• An empty nums list returns false.

Examples

Example 1

Input: ([4, 1, 8, 7], 24)
Output: True

Explanation: (8 - 4) * (7 - 1) = 4 * 6 = 24, so the target is reachable.

Example 2

Input: ([1, 2, 1, 2], 24)
Output: False

Explanation: No combination of 1, 2, 1, 2 with + - * / reaches 24.

Hints

1. Think of the problem as repeatedly replacing two values by the result of one operation, until a single value remains; if that value equals the target (within tolerance) you've found an expression.
2. Try every ordered pair (a, b) so that both a-b and b-a, and a/b and b/a, are explored. Guard division by checking |b| > 1e-6.
3. Use a small epsilon (1e-6) for all equality and division-by-zero checks, because intermediate division can produce non-integers.