Hello,

I've created an example (didn't test) using a Batch class that you can schedule. Here are some considerations about your requirement:

1 - Consider changing the type of field Boolean__c from a picklist to a checkbox. Checkbox field represents a boolean. If not possible, try giving to the field a more meaningful name instead of just "Boolean", this will make more understandable for further reference.
2 - Batch classes are a safe choice when dealing with a high volume of records. Maybe you can do it using a Schedulable interface or a Queueable, is up to you (the query will remain the same).
3 - If this process is for a daily basis use, consider creating an Event Log to help to monitor possible issues.
4 - You can create a metadata to control how often the batch should execute.
5 - There is another approach where you can query the tasks instead, but it would also require another controlling field in the task in order to avoid querying data that was already processed.
 

global class ExampleBatch implements Database.Batchable<sobject> {

    global  Database.QueryLocator start(Database.BatchableContext ctx) {                  
        return Database.getQuerylocator([SELECT Id,
												Boolean__c,
												(SELECT Id FROM Tasks Subject = 'Mass Email: Fourth Followup for Accounts New') 
										FROM 	Contact
										WHERE 	Boolean__c = 'False']); 
    }
    
    global void execute(Database.BatchableContext BC, List<Contact> contacts ) {
    
		Set<Contact> contactsToUpdate = new Set<Contact>();
		
		for (Contact con : contacts) {
			if (con.Tasks.size() > 0) {
				con.Boolean__c = 'True';
				contactsToUpdate.add(con);	
			}			
		}
		
		Database.SaveResult[] saveResults = Database.update(new List<Contact>(contactsToUpdate), false);
		
		// Optional code
		// Iterate through each returned result
		for (Database.SaveResult sr : saveResults) {
			if (sr.isSuccess()) {
				// Operation was successful, so get the ID of the record that was processed
				System.debug('Successfully inserted account. Account ID: ' + sr.getId());
			} else {
				// Operation failed, so get all errors                
				for(Database.Error err : sr.getErrors()) {
					System.debug('The following error has occurred.');                    
					System.debug(err.getStatusCode() + ': ' + err.getMessage());
					System.debug('Contact fields that affected this error: ' + err.getFields());
				}
			}
		}
    }
 
    global void finish(Database.BatchableContext BC) {
		
        // Check batch status - IF COMPLETED then 
        AsyncApexJob job = [SELECT 	Id, Status
                            FROM 	AsyncApexJob 
                            WHERE Id = :bc.getJobId()];        
        
        // Chaining Batch execution
        if (job.Status == 'Completed') {
            
            // If post processing retry is not scheduled then schedule it
            List<CronTrigger> scheduledJobs = [SELECT Id FROM CronTrigger WHERE CronJobDetail.Name = 'ExampleBatch'];
            
            if (scheduledJobs.isEmpty() || Test.isRunningTest()) {

				// You can create a metadata to control how often the batch should execute
				Integer minutes = 30;
				Integer chunkSize = 200;
			
				System.scheduleBatch(new ExampleBatch(), 'ExampleBatch', minutes, chunkSize);
			}
		}
	
    }

}

When you schedule the batch it will reschedule itself (chaining) based on the parameters in the finish method. To start the batch for the first time you can schedule it (System.scheduleBatch(new ExampleBatch(), 'ExampleBatch', 30, chunkSize);) or execute directly (Database.executeBatch(new ExampleBatch());) from the execute anonymous window on the developer console.

Hope to have helped!

Regards.

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