Further enhance this script:

```
import wolframalpha
from g4f.client import Client as GPTClient
import sympy as sp
import mpmath
import tkinter as tk
from tkinter import ttk
import threading
import time
from functools import lru_cache
import random
import re
import numpy as np
from scipy import stats

# Configuration variables
WOLFRAM_APP_ID = 'JG3W9R-WPRHXR7ULG'
GPT_MODEL = 'gpt-4o'
MAX_DEPTH = 20
PRECISION = 100
GPT_MAX_TOKENS = 4069
GPT_TOKENS_PER_MESSAGE = 1024

# Initialize clients
wolfram_client = wolframalpha.Client(WOLFRAM_APP_ID)
gpt_client = GPTClient()

history = []
previous_response = ""

@lru_cache(maxsize=128)
def evaluate_expression(expression):
    try:
        result = sp.sympify(expression)
        return result
    except Exception:
        return None

def high_precision_calculation(expression, precision=PRECISION):
    with mpmath.workdps(precision):
        try:
            result = mpmath.eval(expression)
            return result
        except Exception:
            return None

def estimate_confidence(depth, accuracy, complexity, iterations):
    base_confidence = 0.5 + (depth * 0.05) + (accuracy * 0.2) - (complexity * 0.02) + (iterations * 0.01)
    return round(min(max(base_confidence, 0.0), 1.0), 2)

def calculate_complexity(expression):
    try:
        parsed_expr = sp.sympify(expression)
        complexity = len(str(parsed_expr)) + len(parsed_expr.free_symbols) * 2
        return complexity
    except Exception:
        return 15  # Higher complexity if parsing fails

def advanced_depth_system(problem, max_depth=MAX_DEPTH, max_iterations=10):
    best_result = None
    best_confidence = 0.0
    results = []

    for depth in range(1, max_depth + 1):
        for iteration in range(1, max_iterations + 1):
            time.sleep(0.1 * depth * random.random())
            
            refined_problem = refine_problem(problem, depth)
            potential_solution = f"{refined_problem} (Depth {depth}, Iteration {iteration})"
            accuracy = (depth * 0.05 + iteration * 0.02) * random.uniform(0.9, 1.1)
            complexity = calculate_complexity(problem)
            confidence = estimate_confidence(depth, accuracy, complexity, iteration)

            results.append((potential_solution, confidence))

            if confidence > best_confidence:
                best_result = potential_solution
                best_confidence = confidence

            if confidence > 0.95:
                break

    confidences = [r[1] for r in results]
    mean_confidence = np.mean(confidences)
    std_confidence = np.std(confidences)
    
    z_scores = stats.zscore(confidences)
    outliers = np.abs(z_scores) > 2
    
    filtered_results = [r for r, is_outlier in zip(results, outliers) if not is_outlier]
    
    if filtered_results:
        best_result, best_confidence = max(filtered_results, key=lambda x: x[1])

    return best_result, best_confidence, mean_confidence, std_confidence

def refine_problem(problem, depth):
    try:
        expr = sp.sympify(problem)
        for _ in range(depth):
            expr = sp.simplify(expr)
            expr = sp.expand(expr)
            expr = sp.factor(expr)
            expr = sp.cancel(expr)
            expr = sp.collect(expr, expr.free_symbols)
        return str(expr)
    except Exception:
        return problem

def advanced_auto_translate_to_wolfram(query):
    query = query.lower()
    
    translation_map = {
        'solve': 'compute',
        'find': 'determine',
        'calculate': 'evaluate',
        'simplify': 'simplify the expression',
        'integrate': 'calculate the integral of',
        'differentiate': 'calculate the derivative of',
        'derivative': 'differentiate',
        'integral': 'integrate',
        'limit': 'evaluate the limit of',
        'roots': 'determine the roots of the equation',
        'factor': 'factorize',
        'expand': 'expand the expression',
        'series': 'find the series expansion of',
        'approximate': 'find a numerical approximation of',
        'solve system': 'solve the system of equations',
        'optimize': 'find the optimal value of',
    }
    
    for key, value in translation_map.items():
        query = query.replace(key, value)
    
    return query

def query_wolframalpha(query, max_depth=MAX_DEPTH):
    translated_query = advanced_auto_translate_to_wolfram(query)
    best_result = "WolframAlpha could not find an answer."
    best_confidence = 0.0

    try:
        res = wolfram_client.query(translated_query + ' explain step by step')
        result = next(res.results).text
        best_result, best_confidence, mean_confidence, std_confidence = advanced_depth_system(result, max_depth=max_depth)
        analytical_result = wolfram_to_analytical(best_result)
    except Exception:
        analytical_result = "Could not perform analytical analysis."
        mean_confidence = std_confidence = 0.0

    return analytical_result, best_confidence, mean_confidence, std_confidence

def wolfram_to_analytical(wolfram_result):
    steps = wolfram_result.split('\n')
    analytical_steps = []

    for step in steps:
        step = re.sub(r'\d+(\.\d+)?', 'x', step)
        step = step.replace('Solve[', 'Solve the equation: ')
        step = step.replace('Integrate[', 'Integrate the function: ')
        step = step.replace('Differentiate[', 'Differentiate the function: ')
        analytical_steps.append(step)

    analytical_steps.extend([
        "To solve this problem analytically:",
        "1. Identify the type of problem (e.g., algebraic, calculus, etc.)",
        "2. Choose appropriate mathematical techniques",
        "3. Apply relevant theorems or formulas",
        "4. Simplify and solve step by step",
        "5. Verify the solution",
        "6. Interpret the results in the context of the original problem"
    ])

    return "\n".join(analytical_steps)

def query_gpt(message, suggestion=None):
    global previous_response
    prompt = generate_custom_prompt(message, suggestion)

    response = gpt_client.chat.completions.create(
        model=GPT_MODEL,
        messages=[{"role": "user", "content": prompt}],
        max_tokens=GPT_TOKENS_PER_MESSAGE
    )

    result = response.choices[0].message.content
    previous_response = result

    return result

def generate_custom_prompt(user_input: str, suggestion: str = None) -> str:
    base_prompt = f"""You are an advanced math-solving AI. Your task is to solve the following problem:

Problem: {user_input}

Please provide a detailed, step-by-step explanation of how to solve this problem.
Explain any advanced functions used, and ensure that the solution is accurate and comprehensible. 
Provide the full solution and the final answer.

Chain of thought:
1. Analyze the problem and identify key components.
2. Determine the most appropriate mathematical approach.
3. Break down the solution into logical steps.
4. Apply relevant formulas and techniques.
5. Simplify and refine the solution.
6. Verify the result and provide a clear final answer.
7. Discuss any limitations or assumptions made in the solution.
8. If applicable, suggest alternative approaches or generalizations.

If you need numerical solutions or results, please use the following analytical approach:
"""

    if suggestion:
        base_prompt += f"\n\n{suggestion}"
    else:
        base_prompt += "\n\nNo analytical approach available. Please proceed with your own method."

    return base_prompt

def show_step_by_step_solution(problem):
    try:
        expr = sp.sympify(problem)
        steps = sp.solve(expr, rational=True, steps=True)
        if not steps:
            steps = sp.solve(expr, steps=True)
        return steps
    except Exception:
        return None

def advanced_proof_techniques(problem):
    try:
        expr = sp.sympify(problem)
        simplified = sp.simplify(expr)
        if simplified == expr:
            return "The expression is already in its simplest form."
        else:
            proof_steps = [
                f"Step 1: Original expression = {expr}",
                f"Step 2: Simplify = {simplified}",
                f"Step 3: Expand = {sp.expand(simplified)}",
                f"Step 4: Factor = {sp.factor(simplified)}",
                f"Step 5: Collect terms = {sp.collect(simplified, simplified.free_symbols)}",
                f"Step 6: Cancel common factors = {sp.cancel(simplified)}",
            ]
            
            if expr.has(sp.Symbol('x')):
                limit_inf = sp.limit(expr, sp.Symbol('x'), sp.oo)
                limit_zero = sp.limit(expr, sp.Symbol('x'), 0)
                proof_steps.append(f"Step 7: Limit as x approaches infinity = {limit_inf}")
                proof_steps.append(f"Step 8: Limit as x approaches zero = {limit_zero}")
            
            return "\n".join(proof_steps)
    except Exception:
        return "Could not perform advanced proof."

def chatbot(message):
    wolfram_result, wolfram_confidence, wolfram_mean_conf, wolfram_std_conf = query_wolframalpha(message)

    gpt_result = query_gpt(message, suggestion=wolfram_result)
    gpt_result, gpt_confidence, gpt_mean_conf, gpt_std_conf = advanced_depth_system(gpt_result)
    
    final_result = f"Analytical Approach:\n{wolfram_result}\n\nGPT Solution:\n{gpt_result}\n\nAdvanced Proof:\n{advanced_proof_techniques(message)}"
    confidence = max(wolfram_confidence, gpt_confidence)
    mean_confidence = (wolfram_mean_conf + gpt_mean_conf) / 2
    std_confidence = (wolfram_std_conf + gpt_std_conf) / 2
    
    history.append((message, final_result, confidence, mean_confidence, std_confidence))
    return final_result, confidence, mean_confidence, std_confidence

def on_submit():
    user_input = entry.get()
    submit_button.config(state="disabled")
    threading.Thread(target=process_input, args=(user_input,)).start()

def process_input(user_input):
    response, confidence, mean_conf, std_conf = chatbot(user_input)
    result_text.delete(1.0, tk.END)
    result_text.insert(tk.END, f"Solution: {response}\nConfidence: {confidence}\nMean Confidence: {mean_conf:.2f}\nStd Confidence: {std_conf:.2f}")
    step_by_step_solution = show_step_by_step_solution(user_input)
    if step_by_step_solution:
        result_text.insert(tk.END, f"\n\nDetailed Steps:\n{step_by_step_solution}")
    confidence_label.config(text=f"Confidence: {confidence}")
    submit_button.config(state="normal")

def on_history():
    history_window = tk.Toplevel(root)
    history_window.title("Calculation History")

    history_frame = ttk.Frame(history_window, padding="10")
    history_frame.pack(fill=tk.BOTH, expand=True)

    title_label = ttk.Label(history_frame, text="Calculation History", font=("Helvetica", 14, "bold"))
    title_label.pack(pady=(0, 10))

    history_text = tk.Text(history_frame, height=20, width=60, wrap=tk.WORD)
    history_text.pack(side=tk.LEFT, fill=tk.BOTH, expand=True)

    scroll_bar = ttk.Scrollbar(history_frame, orient=tk.VERTICAL, command=history_text.yview)
    scroll_bar.pack(side=tk.RIGHT, fill=tk.Y)
    history_text.config(yscrollcommand=scroll_bar.set)

    for i, (query, result, confidence, mean_conf, std_conf) in enumerate(history, 1):
        history_text.insert(tk.END, f"{i}. {query} = {result} (Confidence: {confidence}, Mean: {mean_conf:.2f}, Std: {std_conf:.2f})\n")

    history_window.update_idletasks()
    history_window.minsize(history_text.winfo_width() + 50, history_text.winfo_height() + 100)

root = tk.Tk()
root.title("Advanced Math Solver")

style = ttk.Style()
style.theme_use("clam")

entry_frame = ttk.Frame(root, padding="10")
entry_frame.pack(fill=tk.BOTH, expand=True)

entry_label = ttk.Label(entry_frame, text="Enter Math Problem:")
entry_label.grid(row=0, column=0, padx=5, pady=5)

entry = ttk.Entry(entry_frame, width=40)
entry.grid(row=0, column=1, padx=5, pady=5)

submit_button = ttk.Button(entry_frame, text="Submit", command=on_submit)
submit_button.grid(row=1, column=0, columnspan=2, pady=10)

history_button = ttk.Button(entry_frame, text="History", command=on_history)
history_button.grid(row=2, column=0, columnspan=2, pady=5)

result_frame = ttk.Frame(root, padding="10")
result_frame.pack(fill=tk.BOTH, expand=True)

result_text = tk.Text(result_frame, height=10, width=40, wrap=tk.WORD)
result_text.pack(side=tk.LEFT, fill=tk.BOTH, expand=1)
result_scroll_bar = ttk.Scrollbar(result_frame, orient=tk.VERTICAL, command=result_text.yview)
result_scroll_bar.pack(side=tk.RIGHT, fill=tk.Y)
result_text.config(yscrollcommand=result_scroll_bar.set)

confidence_frame = ttk.Frame(root, padding="10")
confidence_frame.pack(fill=tk.BOTH, expand=True)

confidence_label = ttk.Label(confidence_frame, text="Confidence: N/A", font=("Helvetica", 12))
confidence_label.pack(pady=5)

root.mainloop()
```