
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 the sequence $\{a_n\}$ that satisfies $(a_1=a)$, $(a_{n+1}=(2|\sin\frac{n\pi}{2}|-1)a_n+2n$.

(I) Write out the values of $(a_2,a_3,a_4,a_5)$;

(II) Conjecture the general term formula of the sequence $\{a_n\}$, no proof is required;

(III) Using the conjectured conclusion from (II), find the sum of the first 120 terms of the sequence $\{a_n\}$.

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