Paul's priority was the advancement of the gospel, so in every circumstance he looked for a way to spread the gospel message and his love for Jesus. The gospel means good news (an offering of forgiveness/reconciliation from God) in the midst of bad news (condemnation for being enemies of God).
The palace guards were an elite group of soldiers for Rome and Caesar. Paul's character was relatable to these men (courageous, obedient, willing to sacrifice himself for his king). Being in prison was an opportunity to spread the gospel despite his seemingly bad circumstances. This is just like Jesus, wherever He went the gospel advanced - even during His crucifixion. It's not wrong to pray for a change but as long as God has you in that situation, look for how you can share the gospel.
Paul's enemies hoped to produce fear in Christians by imprisoning him but God did the opposite.
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