You are given a high-school competition-level math problem that requires constructing an answer, and its candidate Lean 4 formalization (v4.23.0). Unfortunately, this formalization has error 
and your task is to fix the errors according to the error messages, the retrieved information from documentation and feedback
Your output must follow this template:

1. Start with imports and namespaces such as import Mathlib and open Real.

2. Define the ground-truth answer as an abbrev: 
   `abbrev <problem_name>_answer : β := <expression>`, where β is the type for answer variable. 
   In case the answer depends on context variables, use a lambda function over the context type:
   `abbrev <problem_name>_answer : α → β := fun x => <expression>` Here, α is the type for context variable.
   Do not include input variable bindings (like `(n : ℕ)`) before `: α → β` — instead encode those inside the function body `fun x => <expression>`

3. Write the formal theorem statement using this template or similar way, and leave the proof of the theorem statement by sorry placeholder:
   theorem <problem_name> (x : α) (hypotheses about x) (y : β): P(x, y) ↔ y = <problem_name>_answer x := by sorry

Other Instructions:
- Output your lean code by using delimiter <<<your lean code>>>. 

Example：

Problem:
Given a point $(a,b)$ with $0<b<a$, determine the minimum perimeter of a triangle with one vertex at $(a,b)$, one on the $x$-axis, and one on the line $y=x$.  You may assume that a triangle of minimum perimeter exists.
Answer:
M(a,b) = $\sqrt{2a² + 2b²}$

Formalization:
<<<
import Mathlib

open Set Function Metric

abbrev putnam_1998_b2_answer : ℝ → ℝ → ℝ := 
   fun a b => Real.sqrt (2*a^2 + 2*b^2)

theorem putnam_1998_b2
(a b : ℝ)
(hab : 0 < b ∧ b < a)
: sInf {d : ℝ | ∃ (c : ℝ) (x : ℝ), d = Real.sqrt ((a - c)^2 + (b - 0)^2) +  Real.sqrt ((c - x)^2 + (0 - x)^2) + Real.sqrt ((a - x)^2 + (b - x)^2) ∧
  Real.sqrt ((a - c)^2 + (b - 0)^2) + Real.sqrt ((c - x)^2 + (0 - x)^2) > Real.sqrt ((a - x)^2 + (b - x)^2) ∧
  Real.sqrt ((a - c)^2 + (b - 0)^2) + Real.sqrt ((a - x)^2 + (b - x)^2) > Real.sqrt ((c - x)^2 + (0 - x)^2) ∧
  Real.sqrt ((c - x)^2 + (0 - x)^2) + Real.sqrt ((a - x)^2 + (b - x)^2) > Real.sqrt ((a - c)^2 + (b - 0)^2)}
 = putnam_1998_b2_answer a b := by sorry
>>>
