Brute force is the best way
Play until find the 1274. Unless you find the 728 in the first, second, or third day

This is as Multi-armed Bandit problem. Based on nature of the rewards I think that choosing explore as most often is optimal.
An optimal outcome is hitting 1274 each time. Regret is the difference between optimal outcome and chosen path.
After calculating the average regret for each experiment 1000 times with Epsilon values from .1 to 1--meaning what percent of the time will the algo explore vs exploit its known rewards--my conclusion is explore until finding the 1274 is optimal.

import pandas as pd
import numpy as np
import random
import statistics as sts

def explore():
    reward = random.choice(reward_list)
    known_list.append(reward)
    return max(known_list)

def exploit():
    return max(known_list)

def calculate_regret(results):
    opt = np.full(shape=14, fill_value=1274)  # optimal strategy = 17836
    r = sum(opt) - sum(results)
    print(f'Regret: {r}')
    return r


def run_experiment():    
    results = []
    for j in range(14):
        if random.random() > epsilon:
            results.append(exploit())
        else:
            results.append(explore())
    return results

r_list = []
for i in range(1000):
    # E % of days: explore
    # 1 - E of days:  exploit
    epsilon = 1
    reward_list = [6, 9, 15, 26, 45, 78, 416, 728, 1274]
    known_list = [136]

    results = run_experiment()
    r = calculate_regret(results)
    r_list.append(r)

sts.mean(r_list)

+-----+-------------------+------------------+----------+-----------------+-----------------+
| day | worst brute force | Best brute force | Find 728 | Find 728 2 day  | Find 728 3 day  |
+-----+-------------------+------------------+----------+-----------------+-----------------+
| 1   | 6                 | 1274             | 728      | 416             | 78              |
| 2   | 9                 | 1274             | 728      | 728             | 416             |
| 3   | 15                | 1274             | 728      | 728             | 728             |
| 4   | 26                | 1274             | 728      | 728             | 728             |
| 5   | 45                | 1274             | 728      | 728             | 728             |
| 6   | 78                | 1274             | 728      | 728             | 728             |
| 7   | 416               | 1274             | 728      | 728             | 728             |
| 8   | 728               | 1274             | 728      | 728             | 728             |
| 9   | 1274              | 1274             | 728      | 728             | 728             |
| 10  | 1274              | 1274             | 728      | 728             | 728             |
| 11  | 1274              | 1274             | 728      | 728             | 728             |
| 12  | 1274              | 1274             | 728      | 728             | 728             |
| 13  | 1274              | 1274             | 728      | 728             | 728             |
| 14  | 1274              | 1274             | 728      | 728             | 728             |
| sum | 8967              | 17836            | 10192    | 9880            | 9230            |
+-----+-------------------+------------------+----------+-----------------+-----------------+