
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 that $\{a_{n}\}$ is an arithmetic sequence with $a_{1}=1$, $S_{5}=25$, and $S_{n}$ is the sum of the first $n$ terms of the sequence,  
$(1)$ Find the general formula for the sequence $\{a_{n}\}$;  
$(2)$ If ${c}_{n}=\frac{1}{{a}_{n}{a}_{n+1}}$, find the sum of the first $n$ terms of the sequence $\{c_{n}\}$ denoted as $T_{n}$.

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