
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
Given a sequence $\{a_n\}$ whose sum of the first $n$ terms is $S_n = n^2 + pn$, and a sequence $\{b_n\}$ whose sum of the first $n$ terms is $T_n = 3n^2 - 2n$.

(1) If $a_{10} = b_{10}$, find the value of $p$.

(2) Form a new sequence $\{c_n\}$ by taking the 1st, 3rd, 5th, ... terms of the sequence $\{b_n\}$. Find the general formula for the sequence $\{c_n\}$.

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