# Math Problem Solving - MCTS Example

This example demonstrates using MCTS to solve a math problem by exploring different solution strategies.

```julia
using LLMMCTS

# State represents the current state of problem solving
# It contains the problem statement and the steps taken so far

function math_problem_transition(state::Dict, args::NamedTuple)
    current_step = get(state, :step, 0)
    problem = state[:problem]
    
    # Example problem: Solve x^2 = 16
    if current_step == 0
        # First step: analyze the problem
        newstate = Dict(
            :step => 1,
            :thought => "This is a quadratic equation x^2 = 16",
            :action => "Take square root of both sides",
            :reward => 2.0,
            :isterminal => false
        )
        progressvalue = 5.0
    elseif current_step == 1
        # Second step: solve
        newstate = Dict(
            :step => 2,
            :thought => "Taking square root gives x = ±4",
            :action => "x = sqrt(16) or x = -sqrt(16)",
            :reward => 3.0,
            :isterminal => false
        )
        progressvalue = 7.0
    elseif current_step == 2
        # Third step: verify
        newstate = Dict(
            :step => 3,
            :thought => "Verify both solutions work",
            :action => "Check x=4: 4^2=16 ✓, Check x=-4: (-4)^2=16 ✓",
            :reward => 5.0,
            :isterminal => true  # Problem solved!
        )
        progressvalue = 10.0
    else
        # Terminal state
        newstate = Dict(
            :step => current_step,
            :thought => "Problem solved",
            :action => "Solution complete",
            :reward => 10.0,
            :isterminal => true
        )
        progressvalue = 10.0
    end
    
    return Dict(
        :newNodeKey => "step_$current_step",
        :newstate => newstate,
        :progressvalue => progressvalue
    )
end

# Initial state
initialstate = Dict(
    :step => 0,
    :problem => "Solve x^2 = 16",
    :reward => 0,
    :isterminal => false
)

# Transition arguments
transitionargs = ()

# Run MCTS
result = runMCTS(
    initialstate,
    math_problem_transition,
    transitionargs;
    maxiterations = 15,
    explorationweight = 1.0,
    maxSimulationDepth = 3,
    horizontalSampleExpansionPhase = 3
)

# Display results
println("Problem: ", initialstate[:problem])
println()
println("Best solution trajectory:")
println("  Step ", result.bestTerminalState[:step])
println("  Thought: ", result.bestTerminalState[:thought])
println("  Action: ", result.bestTerminalState[:action])
println()
println("Solution complete! ✓")
println("Root node visits: ", result.root.visits)
```