
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
Let the sequence $\{a_n\}$ satisfy: $a_1=5$, $a_{n+1}+4a_n=5$ $(n\in\mathbb{N}^*)$.  
(1) Is there a real number $t$ such that $\{a_n+t\}$ is a geometric sequence?  
(2) Let $b_n=|a_n|$, find the sum $S_{2013}$ of the first 2013 terms of $\{b_n\}$.

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