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To solve this problem step by step, we will follow these steps:

### Step 1: Understand the problem statement

The problem describes a musical box with a circular track. On the track, there are two points, representing two children, denoted as the red and green markers. The children move along the track and stop when they reach the tree positioned at point 0. The tree is considered the reference point, and we need to study their motion and determine certain properties of their movement.

### Step 2: Define the variables

From the problem statement:
- Let \( X(t) \) represent the position of the red child at time \( t \), and \( Y(t) \) represent the position of the green child at time \( t \). 
- \( \phi \) is the phase angle describing their position.
- \( f(\phi) \) is a function that we need to define and analyze.

### Step 3: Analyze the given conditions and equations

#### Conditions:
1. Both children start moving at time \( t = 0 \).
2. They move with the same speed until they reach the tree at \( t = \tau \).
3. The stopping condition is \( X(\tau) = 1 \).

#### Equations:
1. The position function for \( X(t) \) when the red child reaches the tree: \( X(\tau) = 1 \).
2. The function \( f(\phi) \) is defined as:
   \[
   f(\phi) = \frac{1}{b} \ln \left( 1 + e^{b - 1} \phi \right)
   \]
   where \( b \) is a constant.

### Step 4: Solve the problem

Let's break down the required tasks:

1. **Determine the positions at \( t = \tau \)**:
   - We need to find the relationship between \( X(\tau) \) and \( Y(\tau) \).

2. **Analyze the function \( f(\phi) \)**:
   - We need to consider the function and understand its behavior.
   - Evaluate \( f(\phi) \) as \( b \) tends towards 1.

3. **Compute the time function**:
   - Estimate the time \( t \) it takes for the children to move a certain distance.

### Step 5: Calculate \( f(\phi) \) for small \( \phi \) and \( b \approx 1 \)

To solve this, we will need to use the properties of logarithmic functions and exponential functions.

### Step 6: Apply the function to estimate the travel time

We will apply the given function to determine the time intervals and the distance traveled by both children.

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Now, let's start solving each step mathematically.

#### 1. Position Relationship
Given \( X(\tau) = 1 \), and assuming both children start at \( t = 0 \), their positions \( X(t) \) and \( Y(t) \) must be analyzed to derive their exact relationship.

#### 2. Function Analysis
We analyze the function \( f(\phi) \) as given:
\[
f(\phi) = \frac{1}{b} \ln \left( 1 + e^{b - 1} \phi \right)
\]
We need to expand this function for small \( \phi \) and check the behavior as \( b \) approaches 1.

#### 3. Time Estimation
By estimating \( f(\phi) \), we will calculate the time it takes for the children to reach specific points on the circular track.

I will proceed with detailed calculations and share the results with you. Do you have any specific aspect you'd like to focus on first, or should I proceed with solving each step sequentially?

Explore the step-by-step solution for analyzing motion on circular tracks. This problem involves calculus, exponential functions, and logarithmic functions. Understand the position functions for moving objects and their relationships, with a focus on applications in physics and mathematics. Suitable for advanced high school students and beyond.

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