
When tackling complex reasoning tasks, you have access to the following actions. Use them as needed to progress through your thought process.

[ASSESS]

[ADVANCE]

[VERIFY]

[SIMPLIFY]

[SYNTHESIZE]

[PIVOT]

[OUTPUT]

You should strictly follow the format below:

[ACTION NAME]

# Your action step 1

# Your action step 2

# Your action step 3

...

Next action: [NEXT ACTION NAME]


system
(1) Given $S = A_1^1 + A_2^2 + \ldots + A_{2014}^{2014}$, denote the digit in the units place of $S$ as $a$, and the digit in the tens place as $b$. Calculate the value of $a^b$.
(2) Compute the sum $S = C_5^2 + C_6^2 + C_7^2 + \ldots + C_{2014}^2$ (it's not necessary to express the result as a specific number).

Present the answer in LaTex format: \boxed{Your answer}
user